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Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities A necessary condition for a normal surface singularity to be Gorenstein is that the cycle, obtained by solving the adjunction equations, has integral coefficients. The author proves that this condition (numerically Gorenstein), which depends only on the topological type, is sufficient for the existence of Gorenstein singularity of the same topological type. A resolution of the singularity is constructed by holomorphically plumbing total spaces of line bundles over curves, in such a way that the natural existing two forms extend. The constructed space has the special property that each irreducible component of the exceptional divisor has a neighbourhood isomorphic to a neighbourhood of the zero section of its normal bundle. The result is extended to the \(\mathbb{Q}\)-Gorenstein case: any normal surface singularity is homeomorphic to a \(\mathbb{Q}\)-Gorenstein singularity. Gorenstein singularity; numerical Gorenstein; plumbing Popescu-Pampu, Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones, Duke Math. J. 159 pp 539-- (2011) Singularities in algebraic geometry, Complex surface and hypersurface singularities, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The authors work over the field \(\mathbb C\). An elliptic quotient singularity is a normal two-dimensional singularity which has discrepancy \((-1)\), and its canonical covering is a simple elliptic singularity. A list of these singularities can be found in [\textit{T. S. Gustavsen} and \textit{R. Ile}, Ann. Inst. Fourier 60, No. 2, 389--416 (2010; Zbl 1203.13012)]. Among them, there are only three which admit a smoothing whose Milnor fiber has second Betti number equals to zero [\textit{J. Wahl}, Topology 20, 219--246 (1981; Zbl 1288.32040); Geom. Topol. 15, No. 2, 1125--1156 (2011; Zbl 1220.14003)]. These three elliptic quotient singularities and Wahl singularities (i.e., cyclic quotient singularities \(\frac{1}{n^2}(1,na-1)\) with \(\mathrm{gcd}(n,a)=1\)) form the complete list of log canonical singularities which have a rational homology disk smoothing [\textit{J. Wahl,} Mich. Math. J. 62, No. 3, 475--489 (2013; Zbl 1288.32040)]. Let \((P\in X)\) be one of the three elliptic quotient singularities mentioned above. Then there is a minimal resolution \(\pi:\widetilde X\to X\) such that the exceptional divisor consists of \(4\) smooth rational curves \(E_1\), \(E_2\), \(E_3\), \(F\). The curves \(E_1\), \(E_2\), \(E_3\) are disjoint, each meets the central curve \(F\) transversally at one point, and \([-E_1^2, -E_2^2,-E_3^2;-F^2]=[3,3,3;4], [4,2,4;3]\) or \([2,3,6;2]\). The authors denote the set of these three singularities by \(\mathbb QEq\). The authors use a slightly modified construction of \textit{Y. Lee} and \textit{J. Park} [Invent. Math. 170, No. 3, 483--505 (2007; Zbl 1126.14049)] in order to construct normal projective surfaces \(X\) with invariants \(p_g(X)=0\), \(\pi_1(X)=1\) and \(K^2_X=1,2\) having \(\mathbb QEq\) singularities. In the last section the authors list old and new Wahl singularities having invariants \(p_g=1\), \(\pi_1=1\) and \(K^2=1,2,3,4\). elliptic quotient singularities; KSBA surfaces; Wahl singularities; log canonical singularities Stern, A.; Urzúa, G., KSBA surfaces with elliptic quotient singularities, \(\pi_1 = 1\), \(p_g = 0\), and \(K^2 = 1, 2\), Israel J. Math., 214, 2, 651-673, (2016) Surfaces of general type, Families, moduli, classification: algebraic theory, Singularities of surfaces or higher-dimensional varieties KSBA surfaces with elliptic quotient singularities, \(\pi_{1}=1\), \(p_{g}=0\), and \(K^{2}=1,2\)

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities In a recent paper [the first and the second author, in: Symmetries, integrable systems and representations. Proceedings of the conference on infinite analysis: frontier of integrability, Tokyo, Japan, July 25--29, 2011 and the conference on symmetries, integrable systems and representations, Lyon, France, December 13--16, 2011. London: Springer. 15--33 (2013; Zbl 1307.14007)], the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper, we give a very short geometrical proof of that formula. Qian, C J, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems, 4708-4715, (2005) Parametrization (Chow and Hilbert schemes), (Co)homology theory in algebraic geometry A simple proof of the formula for the Betti numbers of the quasihomogeneous Hilbert schemes

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let C be any reduced and irreducible curve, lying on a smooth cubic surface \(S\subset {\mathbb{P}}^ 3\). In this paper we determine the Hilbert function of C. Moreover we characterize some kinds of curves on S: the arithmetically Cohen-Macaulay curves, the maximal rank curves and the extremal ones. space curve on a smooth cubic surface; maximal rank curves; extremal curves; Hilbert function DOI: 10.1007/BF01762395 Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry, Cycles and subschemes The Hilbert function of a curve lying on a smooth cubic surface

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \((X,0)\) be the germ of an isolated singularity and \(\pi:Y\longrightarrow X\) be a resolution with exceptional divisors \(E_1,\ldots,E_n\). The singularity is called terminal if \(mK_X\) is a Cartier divisor for some integer \(m>0\) and \(K_Y \equiv f^*K_X + \sum a_iE_i, a_i>0\) for all i. The finite subgroups \(G \subset \text{Gl}_4(\mathbb C)\) are classified such that \(\mathbb C^4/G\) has only terminal Gorenstein singularities. action of a finite subgroup; Klein group; irreducible representation Anno, R., Four-dimensional terminal Gorenstein quotient singularities, Math. Notes, 73, 769, (2003) Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays), Singularities of surfaces or higher-dimensional varieties Four-dimensional terminal Gorenstein quotient singularities.

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(k\) be an algebraically closed field of positive characteristic \(p\). If \(g \in k[x_0,x_1,x_2]\) is an irreducible homogeneous polynomial of degree \(d,\) then \(X = \text{ Proj } k[x_0,x_1,x_2]/{(g)}\) is a plane curve of degree \(d\), and there exists the natural embedding \(\eta : X\rightarrow{\mathbb{P}}^2\). Set \(\mathcal M = \eta^*\mathcal{O}_{\mathbb{P}^2}(1)\). Denote by \(\mathrm{HKM}(X,\mathcal M)\) the Hilbert-Kunz multiplicity (HKM) of \(X\) with respect to \(\mathcal M\). The paper proves the conjecture of P. Monsky [personal communication] that if \(X\) is nonsingular with a singular point of multiplicity \(r \geq d/2\), then \(\mathrm{HKM}(X,\mathcal M) = 3d/4 + (2r-d)^2/4d\). Another part of this paper shows that if \(S\) is a two-dimensional standard graded domain over \(k\), of multiplicity \(d\), and \(I\) is a homogeneous ideal generated by a minimal set of generators of degrees \(d_1,\dots,d_t\), then \[ \mathrm{HKM}(S,I) \geq d/2((\sum_{i=0}^td_i)^2/(t-1)- \sum_{i=0}^td_i^2). \] Let \(X =\mathrm{Proj}\,S\to{\mathbb{P}}^N\) be the tautological embedding and \(\pi: X^{\sim} \to X \) the normalization of \(X\). Then the above equality holds iff \(\pi^*K_I\) is a strongly semistable vector bundle on \( X^{\sim}\). In particular, if \(X\) is nonsingular, the equality holds iff \(K_I\) is a strongly semistable vector bundle on \(X\). The proof of this result is based on the following proposition of the paper: Let \(Y\) be a nonsingular curve over \(k\), \(V\) be a rank \(r\) vector bundle of negative degree on \(Y\) and let \(\mathcal L\) be a line bundle of degree \(d>0\) on \(Y\). Then there exists \[ \beta(V) =\lim_{m\rightarrow\infty} 1/p^{2m} \sum_{n=0}^{\lfloor -p^m\mu(V)/d \rfloor -1} h^0(Y, (F^{m*}V)\otimes \mathcal L^{\otimes n}) \] and \(\beta(V) = 0\) iff \(V\) is strongly semistable on \(Y\). Trivedi, V.: Strong semistability and Hilbert -- kunz multiplicity for singular plane curves. Contemp. math. 390, 165-173 (2005) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Plane and space curves Strong semistability and Hilbert-Kunz multiplicity for singular plane curves

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities [For the entire collection see Zbl 0517.00008.] This is another version of an earlier paper of the author [Proc. Symp. Pure Math., Vol. 40, Part II, 479-484 (1983; Zbl 0524.57016)]. normal Gorenstein singularity; cobordism invariants; cobordism group of stably framed 3-manifolds; e-invariant; resolution of singularities; complex analytic surface; Milnor number Algebraic topology on manifolds and differential topology, Singularities of surfaces or higher-dimensional varieties, Complex singularities, Local complex singularities, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory, Stable homotopy of spheres Singularities of complex surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities This book is a classical reference on resolution of algebraic surfaces, embedded into a non-singular projective 3-fold over a perfect field \(k\). It gives a self-contained exposition on results of Zariski and leads up to the author's extensions to the case of characteristic \(p>0\). The principal results are global resolution, global principalization, dominance, birational invariance of dimension of homology groups, as well as (for \(p\neq 2,3,5)\) uniformization and birational resolution. -- The first edition of this book appeared in 1966 (see Zbl 0147.20504). The second edition under review comes in a time of newly increasing interest in resolution of singularities [cf. the work of \textit{A. J. de Jong}, Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996) with an alternative approach, called ``alterations'']. For the recent generation of mathematicians, there may arise some difficulties with the terminology, like the direction of arrows in the introduction (the author is ``blowing down'' where others are ``blowing up''). Here are some remarks on the new edition: It contains an appendix on analytic desingularization in characteristic zero, where a recent short proof is presented. -- From the author's abstract: ``It is hoped that this will remove the fear of desingularization from young minds and embolden them to study it further.'' The proof ``was inspired by discussion with the control theorist Hector Sussmann, the subanalytic geometer Adam Parusiński, and the algebraic geometer Wolfgang Seiler. Once again this illustrates the fundamental unity of all mathematics \dots By an inductive procedure incorporating the principalization lemma, the hypersurface \(f\) is approximated by a binomial hypersurface, i.e. a hypersurface of the form \(X_1^e+ X_2^{b_{e2}}\cdot\dots\cdot X_n^{b_{en}}\), where \(e\) is a positive integer and \(b_{e2},\dots, b_{en}\) are nonnegative integers. The reduction lemma enables us to further arrange matters so that \(\overline{b}_{e2}+\dots+ \overline{b}_{en}<e\), where \(\overline{b}_{e2},\dots, \overline{b}_{en}\) are the residues of \(b_{e2},\dots, b_{en}\) modulo \(e\). This then is a prototype of a good point of a surface.'' The final step is the reduction of multiplicity for such points using a ``good point lemma''. -- The book concludes with an extended bibliography, including also \textit{H. Hauser}'s recent article on ``Seventeen obstacles for resolution of singularities'' [in: Singularities. The Brieskorn anniversary volume, Prog. Math. 162, 289-313 (1998)] which the reader may wish to consult for further developments. resolution of algebraic surfaces; resolution of singularities; analytic desingularization S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monogr. Math., Springer, Berlin 1998. Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Rational and birational maps, Modifications; resolution of singularities (complex-analytic aspects) Resolution of singularities of embedded algebraic surfaces.

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities A review about the classifiction of isolated quotient singularities over \(\mathbb{C}\), i.e. singularities of the form \(\mathbb{C}^N/G\), \(G\subseteq \text{ GL}_N(\mathbb{C})\) a finite subgroup, is given. This is the basis of the description of the Gorenstein isolated quotient singularities. This includes a generalization of the following theorem of Kurano and Nishi: Let \(G\) be a finite subgroup of GL\(_p (\mathbb{C})\), \(p>2\) prime and \(\mathbb{C}^p/G\) an isolated Gorenstein singularity. Then \(\mathbb{V}^p/G\) is a cyclic quotient singularity. Gorenstein; quotient singularity Stepanov, D. A., Gorenstein isolated quotient singularities over \(\mathbb{C}\), Proc. Edinb. Math. Soc. (2), 57, 3, 811-839, (2014) Singularities in algebraic geometry, Local complex singularities Gorenstein isolated quotient singularities over \(\mathbb{C}\)

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Recall that the operad of framed little \(n\)-discs \(fD_n\) consists of configurations of little \(n\)-discs, which are embedded in a fixed unit disc by using a combination of translations, dilatations, and rotations (as opposed to the classical little \(n\)-discs operad of Boardman-Vogt which does not involve any rotation). The homology of the operad of framed little \(2\)-discs is identified with an operad in graded modules governing the category of Batalin-Vilkovsky algebras (see [\textit{E. Getzler}, Commun. Math. Phys. 159, No. 2, 265--285 (1994; Zbl 0807.17026)]). This operad is usually denoted by~\(BV\) in the literature. The authors of the paper under review and P. Ševera proved by different methods that the operad of framed little \(2\)-discs is formal in the sense that the operad of singular chains \(C^{sing}_*(fD_2)\) associated to this operad in topological spaces \(fD_2\) is connected to the corresponding homology operad \(BV = H_*(fD_2)\) by a chain of quasi-isomorphisms of operads in chain complexes \(C^{sing}_*(fD_2)\overset{\sim}\leftarrow\cdot\overset{\sim}{\rightarrow} H_*(fD_2)\) (see [\textit{J. Giansiracusa} and \textit{P. Salvatore}, Contemp. Math. 519, 115--121 (2010; Zbl 1209.18008)] and [\textit{P. Ševera}, Lett. Math. Phys. 93, No. 1, 29--35 (2010; Zbl 1207.55008)]). But the spaces of framed little \(n\)-discs also inherit an extra cycle action which enables us to swap the outputs and inputs of operations together in this operad \(fD_n\), so that \(fD_n\) actually forms a cyclic operad in the sense of \textit{E. Getzler} and \textit{M. Kapranov} [Conf. Proc. Lect. Notes Geom. Topol. 4, 167--201 (1995; Zbl 0883.18013)]. The goal of the paper under review is to prove that the formality quasi-isomorphisms of the operad of little \(2\)-discs can be enhanced to a chain of formality quasi-isomorphisms of cyclic operads. The authors use a cyclic operad \(f\underline{\mathcal{M}}\), weakly-equivalent to the framed little \(2\)-discs \(fD_2\), introduced by \textit{T. Kimura, J. Stasheff} and \textit{A. A. Voronov} [Commun. Math. Phys. 171, No. 1, 1--25 (1995; Zbl 0844.57039)] and defined by certain real oriented blow-ups of the moduli spaces of genus zero stable surfaces with marked points \(\overline{\mathcal{M}}_{0 n}\). The authors adapt a construction of Kontsevich to define a chain of formality quasi-isomorphisms of cyclic operads for this model \(f\underline{\mathcal{M}}\) of the operad of framed little \(2\)-discs \(fD_2\). cyclic operad; formality; framed little discs; genus zero surfaces DOI: 10.1090/S0002-9947-2012-05553-X Loop space machines and operads in algebraic topology, Families, moduli of curves (analytic), Feynman integrals and graphs; applications of algebraic topology and algebraic geometry, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Cyclic operad formality for compactified moduli spaces of genus zero surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities According to \textit{J. Kollár} and \textit{N. I. Shepherd-Barron} [Invent. Math. 91, No. 2, 299-338 (1988; Zbl 0642.14008)] a P-resolution of a surface singularity \(Y\) is a partial resolution \(\pi \colon Z \to Y\) such that \(K_{Z | Y}\) is relatively ample and \(Z\) has at most T-singularities. In the article under review the case \(Y = {\mathbb C}^2 / \Gamma\) with a finite cyclic group \(\Gamma \subset ({\mathbb C}^*)^2\) is considered in the context of toric geometry. By determining the maximal resolution, the author shows that each P-resolution of \(Y\) is again toric. Moreover, he explicitly determines the fans defining the P-resolutions of \(Y\). P-resolution of a surface singularity; cyclic quotient singularities; toric surfaces; T-singularities K. Altmann, P-resolutions of cyclic quotients from the toric viewpoint, in \(Singularities. The Brieskorn Anniversary Volume. Proceedings of the Conference Dedicated to Egbert Brieskorn on his 60th Birthday, Oberwolfach, July 1996\) (Birkhäuser, Basel, 1998), pp. 241-250 (English) Singularities of surfaces or higher-dimensional varieties, Toric varieties, Newton polyhedra, Okounkov bodies, Homogeneous spaces and generalizations P-resolutions of cyclic quotients from the toric viewpoint

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let Hilb\(^p\) be the Hilbert scheme parametrizing the closed subschemes, with fixed Hilbert polynomial \(p=p(t)\in \mathbb Q[t]\), of the projective space \(\mathbb P_K^n\) over an algebraically closed field \(K\) of characteristic \(0\). For each point \(x\in\) Hilb\(^p\), and any \(i\in \mathbb N\), consider the \(i\)-th cohomological Hilbert function \(h_x^i:\mathbb Z\to \mathbb N\), \(m \to \dim_{k(x)}H^i(\mathbb P_{k(x)}^n, \mathcal I^{(x)}(m))\), where \(k(x)\) and \(\mathcal I^{(x)}\subset \mathcal O_{\mathbb P_{k(x)}^n}\) denote the residue field and the ideal sheaf of the subscheme corresponding to \(x\). Now fix a sequence \((f_i)_{i\in\mathbb N}\) of numerical functions \(f_i:\mathbb Z\to \mathbb N\), and consider the sets \(H^{\geq}:=\{x\in\) Hilb\(^p \mid h_x^i\geq f_i \,\,\forall i\in \mathbb N\}\) and \(H^{=}:=\{x\in\) Hilb\(^p\mid h_x^0=f_0, h_x^i\geq f_i \,\,\forall i\geq 1\}\). Semicontinuity implies that \(H^{\geq}\) is closed in Hilb\(^p\), and that \(H^{=}\) is locally closed. In the case \(f_i\equiv 0\) for \(i\geq 1\), many authors gave proofs that \(H^{\geq}\) and \(H^{=}\) are connected, i.e. that the subsets of Hilb\(^p\) which are defined lower bounding the functions \(h_x^0\) are connected. For instance, \textit{R. Hartshorne} [Publ. Math. Inst. Hautes Études Sci. 29, 5--48 (1966; Zbl 0171.41502)] proved that Hilb\(^p\) is connected (see also \textit{G. Gotzmann} [Comment. Math. Helv. 63, No. 1, 114--149 (1988; Zbl 0656.14004)], \textit{D. Mall} [J. Pure Appl. Algebra 150, No. 2, 175--205 (2000; Zbl 0986.14002)], \textit{I. Peeva} and \textit{M. Stillman} [J. Algebr. Geom. 14, No. 2, 193--211 (2005; Zbl 1078.14007)]). Without assumption on the lower bounding functions \(f_i\), in the paper under review the author proves that \(H^{\geq}\) and \(H^{=}\) are connected. The line of the proof is the following. Let \(S:= K[X_0,\dots,X_n]\) be the polynomial ring, and, for any homogeneous ideal \(\mathfrak a\subset S\) and any \(i\in\mathbb N\), consider the \(i\)-th locally cohomological Hilbert function \(h_{\mathfrak a}^i:\mathbb Z\to \mathbb N\), \(m\to \dim_KH^i_{S_+}(\mathfrak a)_m\), where \(H^i_{S_+}(\mathfrak a)\) denotes the graded \(i\)-th local cohomology module of \(\mathfrak a\) with respect to the irrelevant ideal \(S_+\). And denote by \(h_{\mathfrak a}\) the Hilbert function of \(\mathfrak a\). Via Serre-Grothendieck correspondence, one sees that the set of closed points of \(H^{\geq}\) (\(H^{=}\) resp.) is equal to the set \(\mathbb I^{\geq}\) (\(\mathbb I^{=}\) resp.) of saturated homogeneous ideals \(\mathfrak a\subset S\), with Hilbert polynomial \(q(t)={\binom{t+n}{n}}-p(t)\), such that \(h_{\mathfrak a}\geq f_0\) (\(h_{\mathfrak a}= f_0\) resp.) and \(h_{\mathfrak a}^i\geq f_{i-1}\) for all \(i\geq 2\). So to prove that \(H^{\geq}\) and \(H^{=}\) are connected amounts to prove that \(\mathbb I^{\geq}\) and \(\mathbb I^{=}\) are connected. To this aim it suffices to prove that \(\mathbb I^{\geq}\) and \(\mathbb I^{=}\) are connected by Gröbner deformations, i.e. that for any two ideals \(\mathfrak a\), \(\mathfrak b\) \(\in \mathbb I^{\geq}\) (\(\mathbb I^{=}\) resp.), there exists a sequence of ideals \(\mathfrak a=\mathfrak c_1,\dots,\mathfrak c_r=\mathfrak b\) in \(\mathbb I^{\geq}\) (\(\mathbb I^{=}\) resp.) such that \(\mathfrak c_i\) is the saturation of the initial ideal or of the generic initial ideal of \(\mathfrak c_{i+1}\) with respect to some term order or vice versa for all \(i\in\{1,\dots,r-1\}\). To prove that \(\mathbb I^{\geq}\) and \(\mathbb I^{=}\) are connected by Gröbner deformations, the author makes use of certain ideals constructed by \textit{D. Mall} [loc. cit.]. The author proves that a Mall ideal \(\mathfrak c\) is sequentially Cohen-Macaulay, i.e. \(h^i_{\mathfrak c}=h^i_{\text{Gin}_{\text{rlex}}}\mathfrak c\) for all \(i\in \mathbb N\), where \(\text{Gin}_{\text{rlex}}\mathfrak c\) denotes the generic initial ideal of \(\mathfrak c\) with respect to the homogeneous reverse lexicographic order, and that \(\text{Gin}_{\text{rlex}}\mathfrak c=\text{in}_{\text{rlex}}\mathfrak c\). Combining these properties with the general fact that \(h^i_{\mathfrak a}\leq h^i_{\text{in}\mathfrak a}\) for any homogeneous ideal \(\mathfrak a\), any \(i\in \mathbb N\) and any term order, the author is able to conclude the proof of the quoted connectedness property. Hilbert scheme; local cohomology; Mall ideals; sequentially Cohen-Macaulayness S. Fumasoli, Connectedness of Hilbert scheme strata defined by bounding cohomology, PhD thesis, Universität Zürich (2005). arXiv: math.AC/0509123 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Parametrization (Chow and Hilbert schemes) Hilbert scheme strata defined by bounding cohomology

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The paper consists of an introduction to set the stage and to announce the main result (and some of its shortcomings), six sections and references. The first section gives a brief overview of the construction of abelian schemes over Hilbert modular surfaces. In the second section the main object of study, the fundamental relative motive \(\mathcal V^{p,q}\), is introduced. Its various realizations are discussed. In particular, there is a comparison result between the Betti realization and intersection cohomology. The third section deals with the Betti and de Rham realizations. The fourth section gives an overview of \(L\)-functions. The results of Brylinski-Labesse and Chai are discussed. The exposition is quite detailed and technical. In the fifth section the main result (cf. \textit{infra}) is presented. The proof of this result takes quite some place. As usual an important role is played by Beilinson's Eisenstein map. This is explained in the sixth section. Let \(F/{\mathbb Q}\) be a real quadratic field with ring of integers \(\mathcal O_F\). Let \(G/{\mathbb Q} \hookrightarrow\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\) such that the maps \(G\rightarrow{\mathbb G}_m\), \({\mathbb G}_m\hookrightarrow\text{Res}_ {F/{\mathbb Q}}{\mathbb G}_{m,F}\) and \(\text{det}:\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\rightarrow \text{Res}_{F/{\mathbb Q}}{\mathbb G}_{m,F}\) lead to a Cartesian square. The group \(G\) gives rise to a Shimura variety \(S({\mathbb C})\) with a \(G({\mathbb A}_f)\) action. Let \(V^{p,q}=\text{Sym}^pV_2^{\vee}\otimes\text{Sym}^qV _2^{\vee}\), where \(V_2^{\vee}\) is the dual of the standard representation of \(\text{GL}_2\). It is known that intersection cohomology with respect to the Baily-Borel compactification decomposes into \(G({\mathbb A}_f)\)-isotypic components \(H^2_{\text{ét}}(\pi_f)\) which are \(\text{Gal} (\overline{\mathbb Q}/{\mathbb Q})\)-modules. An important result due to J. L.~Brylinski and J.-P.~Labesse says that, for almost all \(p\), the local \(L\)-functions \(L_p(s,H^2_{\text{ét}}(\pi_f))\) can be computed as special values of some automorphic \(L\)-function. This \(L\)-function has simple zeros at \(s=1-n\), \(p+2\leq n\leq p+q+2\) for certain \(\pi_f\), assuming \(p\geq q>0\). It turns out to be appropriate to work with relative motives. Let \(\mathcal A/S\) be the universal abelian scheme over \(S\). \(\mathcal A\) comes equipped with an \(\mathcal O_F\)-action. As a matter of fact, one is interested in constructing elements in \(H^{p+q+3}_{\mathcal M}(\mathcal A^{p+q},\overline{\mathbb Q}(n))\). A basic role is played by the direct summand \(\mathcal V^{p,q}\) of \(R(\mathcal A^{p+q}/S)\), built as suitable \(\text{Sym}^p\otimes\text{Sym}^q\). In Beilinson's philosophy, for a pure (Chow) motive \(M\) with coefficients in \(\overline{\mathbb Q}\) and \(L\)-function \(L(s,H^i_{ \text{ét}}(M))\), one has, for \(n>i/2+1\), regulator maps \[ r_{\mathcal H}:H^{i+1} _{\mathcal M}(M_{\mathbb Z},{\mathbb Q}(n))\rightarrow H^{i+1}_{\mathcal H}(M\otimes {\mathbb R},{\mathbb R}(n)) \] such that there exist \(\overline{\mathbb Q}\)-subspaces \(\mathcal K(i,n) \subset H^{i+1}_{\mathcal M}(M_{\mathbb Z},{\mathbb Q}(n))\) such that \[ r_{\mathcal H} (\mathcal K(i,n))=L(n,H^i_{\text{ét}}(M))\det\mathcal{DR}(i,n), \] where \(\mathcal{DR} (i,n)\) is Deligne's \(\overline{\mathbb Q}\)-structure. In the underlying situation, write \(r_{\mathcal H}:H^{\bullet}_{!\mathcal M}(\mathcal V^{p,q},{\mathbb Q}(*))\rightarrow H^{\bullet}_{!\mathcal H}(\mathcal V^{p,q}\otimes{\mathbb R},{\mathbb R}(*))\) for Beilinson's regulator map. For \(S=S(\pi)\cup\{\ell\}\), let \({L_S(s,H^2_{\text{ét}} (\pi_f)^{\vee}))=\prod_{p\not\in S}L_p(s,H^2_{\text{ét}}(\pi_f)^{\vee})}\). The main result may be stated as follows: Let \(\pi\in\text{coh} (V^{p,q})\) be stable or \(\varepsilon(\pi_f)=-1\); then there is a \(G({\mathbb A}_f)\) -submodule \(\mathcal K(p,q,n)\subset H^{p+q+3}_{\mathcal M}(\mathcal V^{p,q}, {\mathbb Q}(n))\) for all \(p+2\leq n\leq p+q+2\) and \(p\geq q>0\), such that: (i) \(r_{\mathcal H}(\mathcal K(p,q,n))\subset H^{p+q+3}_{\mathcal H}(\mathcal V^{p,q}\otimes {\mathbb R},{\mathbb R}(n))\); (ii) \(r_{\mathcal H}(\mathcal K(p,q,n))(\pi_f)=L_S(n,H^2_ {\text{ét}}(\pi_f))\cdot\mathcal{DR}(p,q,n)(\pi_f)\) (equality of \(\overline{\mathbb Q}\)-subvector spaces in \({\mathbb R}\otimes_{\mathbb Q}\overline{\mathbb Q}\)). \(K(p,q,n)\) can be explicitly constructed. In (ii) \(\mathcal{DR}(p,q,n)(\pi_f)\) is Deligne's \(\overline{\mathbb Q}\)-structure on absolute Hodge cohomology \(H^{p+q+3}_{!\mathcal H}(\mathcal V^{p,q}\otimes{\mathbb R}, {\mathbb R}(n))(\pi_f)\). higher regulators; Hilbert modular surface; motivic cohomology; absolute Hodge cohomology Kings, G., Higher regulators, Hilbert modular surfaces, and special values of \textit{L}-functions, Duke Math. J., 92, 1, 61-127, (1998) Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) Higher regulators, Hilbert modular surfaces, and special values of \(L\)-functions

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(H\) denote a multigraded Hilbert scheme (i.e., \(H\) parametrizes those ideals of a graded polynomial ring \(K[x_1,\dots,x_n]\) that have a fixed multi graded Hilbert function). The torus \(T = (K^*)^n\) acts naturally on \(H\) induced by its action on \(\mathbb{A}^n\). Then the \(T\)-graph of \(H\) is the graph whose vertices are zero-dimensional orbits of \(T\) while the edges are the one-dimensional orbits of \(T\). The \(T\)-graph encodes a great deal of structural information about \(H\). For example \(H\) is connected if and only if its \(T\)-graph is connected. In general, the vertices of the \(T\)-graph will correspond to monomial ideals in \(H\) and an edge will exist between two such ideals \(I\) and \(J\) if and only if there exists a one-dimensional torus whose orbit closure contains both \(I\) and \(J\). As monomial ideals are essentially purely combinatorial objects one expects to be able to resolve questions about the \(T\)-graph by purely combinatorial means. In practice however this is exceedingly difficult. In this paper, the authors find a necessary combinatorial condition for when two vertices in the \(T\)-graph are connected by an edge. They also apply their results to the interesting case of the Hilbert scheme of points in the plane which allows them to, in this case, resolve a question posed by Altman and Sturmfels by showing the \(T\)-graph depends on the characteristic of the ground field. The paper is thorough and the authors illustrate the technique with some very interesting and explicit examples. It is an excellent resource for anyone interested in the multigraded Hilbert scheme. Hilbert schemes; torus actions; Gröbner bases Parametrization (Chow and Hilbert schemes), Polynomial rings and ideals; rings of integer-valued polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) The T-graph of a multigraded Hilbert scheme

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The author proves a formula for cyclotomic polynomials which was required in the study of the characteristic polynomial of the monodromy of an isolated hypersurface singularity. cyclotomic polynomials; monodromy; isolated hypersurface singularity Polynomials in number theory, Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties Cyclotomic polynomials and singularities of complex hypersurfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(S\) be a surface of general type with at most canonical singularities and let \(R(S,K_{S}) =\bigoplus_l H^0(S,lK_{S})\) be the canonical ring of \(S\). The main results of the paper under review are on the degrees of the generators of \(R(S,K_{S})\) when the surface \(S\) is regular and the rational map \(\varphi\) associated to the canonical divisor \(K_{S}\) is a morphism of degree \(n\) on a surface of minimal degree \(r\). In the proof, one considers a general member \(C\) of \(|K_{S}|\) and reduces the problem to analogous ones on the curve \(C\). The authors prove that, if \(n=2\) and \(r=1\), then the canonical ring of \(S\) is generated by its part of degree \(1\) and one generator in degree \(4\); if \(n \neq 2\) or \(r \neq 1\), then the canonical ring of \(S\) is generated by its part of degree \(1\), \(r(n-2)\) generators in degree \(2\) and \(r-1\) generators in degree \(3\). As an application of these results, the authors also construct new examples of surfaces of general type mapping to a surface of minimal degree. Finally, they get a similar result for the graded ring associated to a big and globally generated line bundle \(B\) on a Calabi-Yau threefold \(X\) with at most canonical singularities when the image of the morphism induced by \(B\) is a variety of minimal degree. surfaces of general type; Calabi-Yau threefolds; covering; varieties of minimal degree; canonical ring Gallego F.J., Purnaprajna B.P. (2003). On the canonical rings of covers of surfaces of minimal degree. Trans. Amer. Math. Soc. 355:2715--2732 Surfaces of general type, Calabi-Yau manifolds (algebro-geometric aspects) On the canonical rings of covers of surfaces of minimal degree

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(R=k[[ X_ 1,\ldots,X_ n]]/\mathfrak a\) be a one-dimensional local analytic ring over a perfect field \(k\). It is still, in the general case, an open question whether torsionfreeness of the universally finite differential module \(D(\frac{R}{k})\) implies regularity of \(R\). The author shows that this is so if \(R\) has maximal Hilbert function. In the proof he develops a formula for the drop of the length of this torsion when going from \(R\) to its first quadratic transform \(R_ 1\). From this it follows that this drop is positive if \(R\) is not regular so that the torsion of \(D(\frac{R}{k})\) must be nonzero. The main step of the proof consists in a computation of the relations in \(R_ 1\) from the relations in \(R\), where the maximality of the Hilbert function plays a decisive role. curve singularity; torsionfreeness of the universally finite differential module; maximal Hilbert function T.Pohl, Torsion des Differentialmoduls von Kurvensingularitäten mit maximaler Hilbertfunktion. To appear. Modules of differentials, Singularities of curves, local rings Torsion of the differential module of singularities of curves with maximal Hilbert function

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The holomorphy conjecture roughly states that Igusa's zeta function associated to a hypersurface and a character is holomorphic on \(\mathbb{C}\) whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over \(\mathbb{C}\) with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by \(B_1\)-facets. Singularities in algebraic geometry, Other character sums and Gauss sums, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) The holomorphy conjecture for nondegenerate surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We study the germs of curves embedded in a rational surface singularity \((S,P)\) from the point of view of proximity. We generalize the geometric theory of Enriques, obtaining necessary and sufficient conditions to impose on a finite set of points infinitely near \(P\) with assigned orders to get effective Weil (resp. Cartier) divisors going through these points with these orders (where, since a Weil divisor \(C\) on \((S,P)\) is \(\mathbb{Q}\)- Cartier, the orders of \(C\) are defined to be the rational numbers determined by the valuations given by the exceptional curves in the respective point blowing ups). We apply this result to give a formula to compute the minimal numbers of generators of any \({\mathfrak m}\)-primary complete ideal \(I\) of \({\mathcal O}_{S,P}\), generalizing the formula given by Hoskin and Deligne in the nonsingular case. We also give an algorithm to describe a minimal system of generators of \(I\). The germs of reduced curves in \((S,P)\) are classical up to a notion of equisingularity which generalizes the equisingularity of germs of plane curves. The equisingularity class of such a germ of curve \(C\) in \((S,P)\) consists of the weighted dual graph of the minimal embedded desingularization of \(C\) in \((S,P)\), together with some weighted arrows corresponding to the branches of \(C\). We describe an algorithmic procedure giving the sequence of multiplicities of the successive strict transforms of each branch of \(C\) from the equisingularity class of \(C\). curves embedded in a rational surface singularity; proximity; divisors; equisingularity Reguera, A. J.: Courbes et proximité sur LES singularités rationnelles de surface. CR acad. Sci. Paris sér. I math. 319, 383-386 (1994) Singularities of surfaces or higher-dimensional varieties, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects), Curves in algebraic geometry, Local deformation theory, Artin approximation, etc. Curves and proximity in rational surface singularities.

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The classification of maximal Cohen-Macaulay (MCM) modules over Noetherian local rings is a difficult problem in general, and it has a long history; we refer the reader to the book [\textit{G. Leuschke} and \textit{R. Wiegand}, Cohen-Macaulay representations. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1252.13001)] for a detailed overview of the subject. Let \(k\) be an algebraically closed field such that \(\text{char}(k) \neq 2\). The goal of the article under review is to study the MCM representation type of rings of the form \[ k[[x,y,z]]/(xy, y^q - z^2), \] and also the ring \[ k[[x,y,z]]/(xy, z^2) \] (additional results in the case \(\text{char}(k) = 2\) are also obtained; see Remark 2.3 of the article under review for details). The main results of the article are as follows. The authors first prove that the above rings have \textit{tame} MCM representation type (see Section 4 of the article for the definition of tameness). They then go on to give an explicit description of all indecomposable MCM modules over the ring \(k[[x,y,z]]/(xy, z^2)\). Finally, the authors apply the previous results to construct explicit families of \textit{matrix factorizations} associated to the hypersurface ring \(k[[x,y]]/(x^2y^2)\). Matrix factorizations were introduced in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]; roughly speaking, they are the data of two-periodic tails of minimal free resolutions of finitely generated modules over hypersurface rings. maximal Cohen-Macaulay modules; matrix factorizations; MCM representation type Burban, I., Gnedin, W.: Cohen-Macaulay modules over some non-reduced curve singularities. arXiv:1301.3305v1 Cohen-Macaulay modules, Cohen-Macaulay modules in associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Singularities in algebraic geometry Cohen-Macaulay modules over some non-reduced curve singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities These notes are based on series of seven lectures given in the combinatorics seminar at U.C. San Diego; in February and March, 2001. We discuss a series of new results in combinatorics, algebra and geometry. The main combinatorial problems we solve are (1) we prove the positivity conjecture for Macdonald polynomials, and (2) we prove a series of conjectures relating the diagonal harmonics to various familiar combinatorial enumerations; in particular we prove that the dimension of the space of diagonal harmonics is \((n+1)^{n-1}\). In order to prove these results, we have to work out some new results about geometry of the Hilbert scheme of points in the plane and a certain related algebraic variety. As a technical tool for our geometric results, in turn, we need to do some commutative algebra, which although complicated, has a quite explicit and combinatorial nature. symmetric functions; positivity conjecture; diagonal harmonics; Hilbert scheme of points in the plane Mark Haiman, Notes on Macdonald polynomials and the geometry of Hilbert schemes, Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, pp. 1 -- 64. Parametrization (Chow and Hilbert schemes), Extremal set theory Notes on MacDonald polynomials and the geometry of Hilbert schemes.

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The paper under review provides one of the first examples of invariant Hilbert schemes with multiplicities introduced in [\textit{V. Alexeev} and \textit{M. Brion}, J. Algebr. Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005)]. \(\text{SL}_2\) acts on \((\mathbb{C}^2)^6\) and the moment map \(\mu\) defines the symplectic reduction \(\mu^{-1}(0)// \text{SL}_2\). The paper under review describes explicitly the invariant Hilbert scheme \(\text{Hilb}^{\text{SL}_2}_h(\mu^{-1}(0))\) for the Hilbert function \[ h:\mathbb{N}_0\to \mathbb{N},\quad d\mapsto d+1, \] and proves it is connected and smooth. The Hilbert-Chow morphism \[ \text{Hilb}^{\text{SL}_2}_h(\mu^{-1}(0))\to \mu^{-1}(0)//\text{SL}_2 \] factors through the well known symplectic resolutions of \(\mu^{-1}(0)//\text{SL}_2\) given by the cotangent bundles of \(\mathbb{P}^3\) and its dual. The method of the paper provides a general procedure of these calculations that can be applied to similar examples. invariant Hilbert schemes T. Becker, \textit{An example of an} SL\_{}\{2\}-\textit{Hilbert scheme with multiplicities}, Transform. Groups \textbf{16} (2011), no. 4, 915-938. Formal groups, \(p\)-divisible groups, Birational geometry, Special varieties An example of an SL\(_{2}\)-Hilbert scheme with multiplicities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(G\) be a finite group acting on a quasi-projective smooth scheme \(X\) over a field \(k\). Suppose that \(g:\tilde{Y}\to X/G\) is a resolution of singularities of \(X/G\). Then by the McKay principal many aspects of the geometry of \(\tilde{Y}\) is reflected in the \(G\)-equivariant geometry of \(X\). Assume that \(E=g^{-1}(Z)_{\text{red}}\) where \(Z \subset X/G\) is the singular locus is a divisor with simple normal crossing. Denote by \(\Gamma(E)\) the dual complex associated to \(E\). For example, in the case of the Klein singularity \(\mathbb{C}^2/G\), \(\Gamma(E)\) is one of the ADE Dynkin diagrams. The paper under review studies the homotopy type of \(\Gamma(E)\) which is known to be independent of the choice of the resolution. To express the result let \(\pi:X\to X/G\) be the projection, \(T=g^{-1}(Z)_{\text{red}}\), and \(f:\tilde{X}\to X\) be a proper birational \(G\)-equivariant morphism such that \(\tilde{X}\) is a smooth \(G\)-scheme and \(E_T=f^{-1}(T)_{\text{red}}\) is a \(G\)-strict simple normal crossing divisor and \(f\) is an isomorphism over \(X-T\). The main result of the paper under review proves that if \(k\) is perfect with characteristic zero, then there is a canonical map \(\phi: \Gamma(E_T)/G\to \Gamma(E)\) in the homotopy category of \(CW\)-complexes which induces isomorphisms on the homology and fundamental groups. This result has important implications such as 1) \(\Gamma(E_T)/G\) is contractible \(\Rightarrow\) \(\Gamma(E)\) is contractible. 2) \(X/G\) have isolated singularities \(\Rightarrow\) T is smooth \(\Rightarrow\) \(\Gamma(E)\) is contractible. The question of the contractibility of \(\Gamma(E)\) is very important, for example if \(X/G\) has rational singularities then \(\Gamma(E)\) is contractible. The proof of the main result regarding the fundamental group uses a geometric interpretation of the fundamental group by means of the classifying group of \(cs\)-coverings of \(E\). And regarding the homology groups the proof is based on an equivariant weight homology theory introduced in the paper under review, and proving an analog of McKay principal for this weight homology. The proof of the latter relies on introducing another (arithmetic) homology theory in the paper under review called equivariant Kato homology. The other ingredients of the proof are cohomological Hasse principal, Deligne's theorem on Weil conjecture and Gabber's refinement of De Jong's alteration theorem. McKay principal; Equivariant geometry Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence, Geometric class field theory Cohomological Hasse principle and resolution of quotient singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper [\textit{I. Cherednik} and \textit{I. Philipp}, Algebr. Geom. Topol. 18, No. 1, 333--385 (2018; Zbl 1396.14026)] from rank one to any ranks; explicit formulas are obtained for torus knots. The new feature is the definition of counterparts of Jacobian factors (directly related to compactified Jacobians) for higher ranks, which is parallel to the classical passage from invertible sheaves to vector bundles over algebraic curves. This is an entirely local theory, connected with affine Springer fibers for non-reduced (germs of) spectral curves. We conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding DAHA superpolynomials for any algebraic knots colored by columns. Hecke algebra; Khovanov-Rozansky homology; algebraic knot; Macdonald polynomial; plane curve singularity; compactified Jacobian; Puiseux expansion; orbital integral Plane and space curves, Root systems, Lie algebras of linear algebraic groups, Hecke algebras and their representations, Braid groups; Artin groups, Representations of Lie and linear algebraic groups over local fields, Geometric Langlands program: representation-theoretic aspects, Compact Riemann surfaces and uniformization, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics, Knot polynomials Modules over plane curve singularities in any ranks and DAHA

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We show how to apply a theorem by Lê Dung Trang and the author about linear families of curves on normal surface singularities to get new results in this area. The main concept used is a precise definition of general elements of an ideal in the local ring of the surface. We make explicit the connection between this notion and the more elementary notion of general element of a linear pencil, through the use of integral closure of ideals . This allows us to prove the invariance of the generic Milnor number (resp. of the multiplicity of the discriminant), between two pencils generating two ideals with the same integral closure (resp. the projections associated). We also show that our theorem, applied in two special cases, on the one hand completes, removing an unnecessary hypothesis, a theorem by J. Snoussi on the limits of tangent hyperplanes, and on the other hand gives an algebraic-constant theorem in linear families of planes curves. Surface singularity; general element; Milnor number; integral closure of ideals; complete ideals; limits of tangent hyperplanes; discriminants Equisingularity (topological and analytic), Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings General elements of an \(m\)-primary ideal on a normal surface singularity

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities ``In the last years many new results on the classical problem of classifying smooth surfaces in the projective space in terms of their extrinsic projective and intrinsic geometric invariants have been made by using the adjunction mapping. We extend the existence theorem for the adjunction mapping to the case of singular surfaces. Although the mapping is only meromorphic we obtain many inequalities, known previously only in the smooth case. As an illustration of the results we give a very complete answer in the singular case, parallel to the smooth result, to the question of when a singular surface can `have a hyperelliptic hyperplane section'''. adjunction mapping; singular surfaces; hyperelliptic hyperplane section Andreatta M., Beltrametti M., Sommese A. J., Generic properties of the adjunction mapping for singular surfaces and applications. Preprint. Divisors, linear systems, invertible sheaves, Singularities of surfaces or higher-dimensional varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Generic properties of the adjunction mapping for singular surfaces and applications

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We present a CR-construction of the versal deformations of the singularities \(V_n\simeq\mathbb{C}^2/\mathbb{Z}_n\), \(n\in\{2,3,4,\dots\}\) defined by the immersions \(X^n:(z,w)\to (z^n,z^{n-1}w,\dots,zw^{n-1},w^n)\) of \(\mathbb{C}^2\) into \(\mathbb{C}^{n+1}\). obstruction Deformations of complex singularities; vanishing cycles, Deformations of special (e.g., CR) structures, Deformations of singularities, Complex surface and hypersurface singularities, CR structures, CR operators, and generalizations, CR manifolds as boundaries of domains CR deformations for rational homogeneous surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Stable envelopes are special classes in the cohomology or \(K\)-theory or elliptic cohomology of geometrically relevant moduli spaces. They govern the enumerative geometry of the space, the associated representation theory of a quantum group, and are related to solutions of some associated differential equations. In all these areas concrete formulas for stable envelopes are wanted. The paper under review gives concrete formulas for the elliptic stable envelopes when the moduli space is the Hilbert scheme of points on the space. Stable envelopes are associated to the torus fixed points of the space. The torus fixed points on the Hilbert scheme are parametrized by certain tuples of partitions. The author describes some trees subordinate to those partitions. To each tree a product of elliptic functions (and their inverses) is assigned, and the formula is a sum of such functions for all trees. The proof technique is abelianization: an appropriate resolution of the attracting set of the fixed point, calculating the relevant class in the resolution, and pushing it forward. As a byproduct \(K\)-theoretic and cohomological stable envelopes are also obtained, as the trigonometric and rational limits of the elliptic stable envelope formula. quiver variety; stable envelope; elliptic cohomology; Hilbert scheme of points in the plane Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial aspects of representation theory, Algebraic moduli problems, moduli of vector bundles, Mirror symmetry (algebro-geometric aspects), Equivariant \(K\)-theory, Analytic spaces Elliptic stable envelope for Hilbert scheme of points in the plane

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(R = {\mathbb C}[x_1,\ldots, x_n]\) be the polynomial ring graded by a non-negative \(d\times n\)-matrix \(A = (a_1, \ldots, a_n)\) of non-negative integers such that \(\deg x_i = a_i \in {\mathbb N}^d\) is given by a vector. This defines a decomposition \(R = \bigoplus_{b \in {\mathbb N}A} R_b,\) where \({\mathbb N}A\) denotes the subsemigroup of \({\mathbb N}^d\) generated by the vectors \(a_1, \ldots, a_n\) and \(R_b\) is the \({\mathbb C}\)-span of an element \(b\) of the subsemigroup. The toric Hilbert scheme parametrizes all the \(A\)-homogeneous ideals \(I \subset R\) with the property that the graded component \((R/I)_b\) is a 1-dimensional \({\mathbb C}\)-vector space. This concept was summarized by \textit{B. Sturmfels} in the first chapter of his book ``Gröbner bases and convex polytopes'', Univ. Lect. Ser. 8 (1996; Zbl 0856.13020)]. In the paper under review, the authors illustrate the use of Macaulay 2 for exploring the structure of toric Hilbert schemes. It is known that all the components of the scheme are toric varieties. Among them there is a fairly well understood component, the coherent component. The results contribute to the open problem whether toric Hilbert schemes are always connected. In their investigations the authors encounter algorithms from commutative algebra, polyhedral theory and geometric combinatorics. Macaulay2; toric Hilbert scheme; semigroup algebra Stillman, M., Sturmfels, B., Thomas, R.: Algorithms for the toric Hilbert scheme. In: Computations in Algebraic Geometry using Macaulay 2, D. Eisenbud et al. (eds.), Algorithms and Computation in Mathematics Vol 8, Springer, 2002, pp. 179--213 Computational aspects of higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to algebraic geometry, Symbolic computation and algebraic computation Algorithms for the toric Hilbert scheme

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities In 1982, \textit{J. Stückrad} and \textit{W. Vogel} [Curves Semin. at Queen's, Vol. 2, Queen's Pap. Pure Appl. Math. 61, 1--32 (1982; Zbl 0599.14003)] introduced the notion of \textit{self-intersection} cycle, for a projective variety \(S\subset \mathbb P^N\), which comes out from the join \(\{\langle x,y\rangle: x,y\in S\}\). The self-intersection cycle has numerical properties which are linked to the main invariants of the embedding of \(S\). In the paper under review, the authors use the self-intersection cycle to find, in any codimension, a formula which links \(\deg (S)\) to the degree of the secant, tangent and polar loci of \(S\), when \(S\) is a smooth surface. This formula is helpful in the study of the ramification locus of general projections of \(S\). When \(S\) is a singular surface, a similar formula holds, but one needs to add a contribution from the singularities of the surface. For general surfaces, it seems difficult to to compute the new summand. The authors show how one can estimate the contribution from the singularities, for a particular class of non-normal, Del Pezzo surfaces. surfaces; projections Achilles, R.; Manaresi, M.; Schenzel, P.: On the self-intersection cycle of surfaces and some classical formulas for their secant varieties, Forum math. 23, 933-960 (2011) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Ramification problems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties On the self-intersection cycle of surfaces and some classical formulas for their secant varieties

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let X be a normal Gorenstein complex surface such that its anticanonical sheaf \(\omega_ X^{-1}\) is ample. If \(d=\omega_ X\cdot \omega_ X\) is the degree of X, then it is known that \(1\leq d\leq 9.\) In this paper complete results for \(2\leq d\leq 9\) (and partial results for \(d=1)\) are obtained concerning the type of singularities such a surface X can have. All corresponding models are obtained either starting from \({\mathbb{P}}^ 2\) and performing certain blowing-ups and blowing-downs (explicitly described in the paper) or by taking a cone over an elliptic curve. The special case \(d=3\) corresponds to cubic surfaces in \({\mathbb{P}}^ 3\) and was treated with completely different methods by \textit{J. W. Bruce} and \textit{C. T. C. Wall} [J. Lond. Math. Soc., II. Ser. 19, 245-256 (1979; Zbl 0393.14007)]. Also the special case \(d=4\) corresponds to complete intersections of type (2,2) in \({\mathbb{P}}^ 4\) and can be derived from general results of \textit{H. Knörrer} [Bonn. Math. Schr. 117 (1980; Zbl 0457.14003)]. classification of singularities; ample anticanonical divisor; normal Gorenstein complex surface; type of singularities O. Păsărescu, On the Classification of Singularities of Normal Gorenstein Surfaces with Ample Anticanonical Divisor (Romanian). Stud. Cerc. Mat.36, 227--243 (1984). Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Special surfaces On the classification of singularities of normal Gorenstein surfaces with ample anticanonical divisor

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Translation from Izv. Akad. Nauk SSSR, Ser. Mat. 45, 398-434 (Russian) (1981; Zbl 0493.14014). Tate conjecture on algebraic cycles; Hodge conjecture; Pic; etale cohomology S.G. Tankeev , On algebraic cycles on surfaces and abelian varieties . Math USSR-Izv., vol. 18(2) (1982) 349-380. Arithmetic problems in algebraic geometry; Diophantine geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Analytic theory of abelian varieties; abelian integrals and differentials, \(p\)-adic cohomology, crystalline cohomology, Picard groups, Transcendental methods of algebraic geometry (complex-analytic aspects) On algebraic cycles on surfaces and abelian varieties

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Der Grundkörper ist \(k=\mathbb{C}\), und \({\mathcal H}:=H_{d,g}= \text{Hilb}^P (\mathbb{P}^3_k)\), \(P(T)= dT-g+1\), ist das (volle) Hilbertschema der Raumkurven vom Grad \(d\) und Geschlecht \(g\). Für \(3\leq d\leq 11\) definiert man \(g(d)\) durch die Tabelle \[ \begin{matrix} d & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11\\ g(d) & -2 & 0 & 1 & 2 & 4 & 6 & 9 & 11 & 15\end{matrix} \] und für \(d\geq 12\) durch die Formel \(g(d)={1\over 6}d(d-3)\). Die vorliegende Arbeit bringt zunächst eine Ergänzung und verschiedene Verbesserungen zu früheren Arbeiten des Autors [\textit{G. Gotzmann}, ``Der kombinatorische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' und ``Der algebraische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' (Münster 1994; Zbl 0834.14004 und Münster 1997; Zbl 0954.14002)], und man erhält als Zusammenfassung: Satz I. Wenn \(d\geq 3\) und \(g\leq g(d)\) ist, dann ist (i) \(\dim_\mathbb{Q} A_1({\mathcal H})\otimes_\mathbb{Z} \mathbb{Q}=3\); (ii) \(\text{Pic} ({\mathcal H}) \simeq \mathbb{Z}^3\oplus \mathbb{C}^r\), mit \(r=\dim_kH^1 ({\mathcal H},{\mathcal O}_{\mathcal H})\); (iii) \(\text{NS}({\mathcal H})\simeq\mathbb{Z}^3\). Der Satz ist eine Folgerung aus etwas genaueren Ergebnissen in Abschnitt 6, wo explizite Basen von \(A_1({\mathcal H})\otimes\mathbb{Q}\) und \(\text{NS}({\mathcal H})\) bestimmt werden. -- Man kann die in Satz I genannten Ergebnisse für \({\mathcal H}\) auf die zugehörige universelle \({\mathcal C}\subset{\mathcal H}\times\mathbb{P}^3\) übertragen, und man erhält: Satz II. Wenn \(d\geq 3\) und \(g\leq g(d)\) ist, dann ist (i) \(\dim_\mathbb{Q} A_1({\mathcal C})\otimes_\mathbb{Z} \mathbb{Q}=4\); (ii) \(\text{Pic}({\mathcal C})\simeq \mathbb{Z}^4\oplus \mathbb{C}^r\), mit \(r=\dim_kH^1 ({\mathcal C},{\mathcal O}_{\mathcal C})\); (iii) \(\text{NS}({\mathcal C})\simeq\mathbb{Z}^4\). Néron-Severi group; Hilbert scheme; universal curve; Picard group Parametrization (Chow and Hilbert schemes), Plane and space curves The Néron-Severi group of a Hilbert scheme of space curves and the universal curve

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The first counterexample to Kodaira vanishing in positive characteristic was constructed by \textit{M. Raynaud} [in: C.P. Ramanujam. - A tribute. Collect. Publ. of C.P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 273--278 (1978; Zbl 0441.14006)], many other counter-examples have been found satisfying various prescribed properties. In this paper, let \(k\) be a field of charactristic 2. The author shows that there exists a Fano variety \(X\) of dimension six over \(k\) such that anticanonical sheaf \(\omega_X^{-1}\) is very ample and \(\omega_X^{-2}\) violates Kodaira vanishing. By taking the cone over \(X\) given the emedding induced by the global sections of \(\omega_X^{-1}\), there exists a variety \(Z\) of dimension \(7\) with a single isolated canoncial singularity over \(k\), but \(Z\) is not Cohen-Macaulay. Fano variety; Kodaira vanishing theorem; positive characteristic Singularities in algebraic geometry, Rational and birational maps, Fano varieties Non-Cohen-Macaulay canonical singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(X\) be a normal projective surface over an algebraically closed field \(k\) of characteristic \(p>0\) and \(x\in X\) be a singular point. \((X,x)\) is called \(F\)-pure if the Frobenius map \(F:{\mathcal O}_{X,p}\to{\mathcal O}_{X,p}\) is pure, a notion due to \textit{M. Hochster} and \textit{J. L. Roberts} [Adv. Math. 21, 117-172 (1976; Zbl 0348.13007)]. Let \(f:Y\to X\) be the minimal resolution of the singularity at \(p\) with \(E\) the reduced exceptional divisor. The authors show the following characterization of \(F\)-pure surface singularities in characteristic \(p\): Let \(E=\bigcup^ n_{i=1}E_ i\) \((E_ i\) irreducible) have only normal crossings. Then \((X,x)\) is \(F\)-pure if and only if one of the conditions (i)--(iv) hold: (i) \(E\) is an irreducible smooth ordinary elliptic curve, (ii) \(E\) is an irreducible rational curve which has only one singularity (which is necessarily nodal), (iii) \(E_ i\cong\mathbb{P}^ 1\) for all \(i\) and the dual graph of \(E\) is an \(n\)-gon, (iv) \(E_ i\cong\mathbb{P}^ 1\) for all \(i\) and the graph of \(E\) is a tree (in particular, \((X,x)\) is rational). Rational double points in characteristic \(p>5\) are \(F\)-pure. Hence, the theorem gives a classification of normal \(F\)-pure surface singularities in characteristic \(p>5\). An analogous result in the Gorenstein case was obtained by \textit{K. I. Watanabe} [Algebraic geometry and commutative algebra, in Honor of M. Nagata, 791-800 (1988)]. \(F\)-pure surface singularities in characteristic \(p\) V. B. Mehta and V. Srinivas, Normal \?-pure surface singularities, J. Algebra 143 (1991), no. 1, 130 -- 143. Singularities of surfaces or higher-dimensional varieties Normal F-pure surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Harder, Langlands and Rapoport had proved the Tate conjecture for algebraic cycles for the non-CM-submotives of \(H^ 2\) of Hilbert modular surfaces [\textit{G. Harder}, \textit{R. P. Langlands}, \textit{M. Rapoport}, J. Reine Angew. Math. 366, 53--120 (1986; Zbl 0575.14004)]. The authors consider Hilbert modular forms of CM-type and prove the Tate conjecture for the CM-submotives in \(H^ 2\). This gives the full Tate conjecture asserting the algebraicity of all Tate cycles on Hilbert modular surfaces over an arbitrary number field. The authors also relate, in the CM case, the number of independent divisor classes to the order of pole of the \(L\)-function of the surface at the edge of convergence. The results have interesting corollaries about the Picard group of the surface. There exist non-trivial divisors defined over metabelian fields which are not in the linear span of the Hirzebruch-Zagier cycles, canonical divisors and the cycles coming from the desingularization of the surface. The methods are quite different from the methods of Harder, Langlands and Rapoport. Hodge cycles; CM-motives; Tate conjecture for algebraic cycles; Hilbert modular surfaces; Hilbert modular forms of CM-type; Picard group V. K. Murty, D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89 (1987), no. 2, 319--345. Cycles and subschemes, Special surfaces, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Generalizations (algebraic spaces, stacks), Picard groups Period relations and the Tate conjecture for Hilbert modular surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(X\) be a real coherent analytic space, \({\mathcal O}_X\) the sheaf of analytic functions, \({\mathcal O}(X)\) the ring of global analytic functions. Given an ideal \({\mathfrak a}\subseteq{\mathcal O}(X)\), the zero set of \({\mathfrak a}\) is denoted by \({\mathcal Z}({\mathfrak a})\), the vanishing ideal of \({\mathcal Z}({\mathfrak a})\) is denoted by \({\mathcal I}({\mathcal Z}({\mathfrak a}))\). Nullstellensätze describe the connections between the ideals \({\mathfrak a}\) and \({\mathcal I}({\mathcal Z}({\mathfrak a}))\): When is it true that \({\mathfrak a}={\mathcal I}({\mathcal Z}({\mathfrak a}))\)? The authors show that, given a real coherent analytic surface \(X\), the equality \({\mathfrak a}={\mathcal I}({\mathcal Z}({\mathfrak a}))\) holds if and only if \({\mathfrak a}\) is a real ideal and is saturated (i.e., \({\mathfrak a}\) is the ideal of global sections of the ideal sheaf generated by \({\mathfrak a}\)). It remains an open question whether the result can be extended to the space \(\mathbb{R}^3\). However, it is shown that the result fails for \(\mathbb{R}^3\) if and only if there is a so-called special irreducible functions that generates a real ideal. Primary ideals and primary decompositions of saturated ideals are among the main tools of the paper. real analytic space; analytic function; zero set; vanishing ideal; Nullstellensatz; primary decomposition Broglia, F; Pieroni, F, The nullstellensatz for real coherent analytic surfaces, Rev. Mat. Iberoam., 25, 781-798, (2009) Real-analytic and semi-analytic sets, Real algebraic and real-analytic geometry, Germs of analytic sets, local parametrization, Sums of squares and representations by other particular quadratic forms The Nullstellensatz for real coherent analytic surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The compound Du Val singular points (cDV points, in short) are expected to play important roles in a theory of minimal models of varieties of dimension 3. The author tried to provide a complete list of their normal forms. This article is a report on his attempt. In fact, he gave some preparatory results and gave a rigorous foundation for Arnold's theory of spectral sequences for reduction of functions. However, he gave only a few examples of normal forms. Because of an immense increase in the amount of elementary computation, he could not give the list. In spite of the very sound frame of his theory, it is strange that we can find a lot of misunderstandings and misprints even in the Russian version of this article. Now, a three-dimensional hypersurface singular point on a complex variety is called a cDV point if the intersection with a general hyperplane passing through the point is a rational double point of dimension 2. A wider class of singular points is called canonical singularities. In section 1 a characterization of hypersurface canonical singularities with respect to the Newton diagrams and the diagonal point (1,1,...,1) is given. In section 2, after a characterization for normal forms of functions with 4 variables to define cDV points, the author begins the preparation to give normal forms for cDV points with a special form. Section 3 is devoted to the explanations of Arnold's theory of spectral sequences for reduction of functions. In section 4 the author carries out the computation for the reduction of functions for a few cases by using spectral sequences. Let us have a look on part I of the proof of theorem 8, for example. Here the author has forgotten to treat the monomial \(y^ 2z^{\ell}\) appearing in \(\pi\) when \(2\alpha_ 1\geq \alpha_ 2>\alpha_ 1\) and \(\alpha_ 3| 2\alpha_ 1-\alpha_ 2\). Thus theorem 8 is not verified. Moreover, in the proof of lemma 2' in section 3, the Taylor expansion is strange, and we cannot understand why the author uses condition A but does not use the following coordinate change associated with the vector field \(s=\sum v_ i\partial /\partial x_ i:\) \((*)\quad y_ i=x_ i+\sum^{\infty}_{j=1}(j!)^{-1}s^ j(x_ i)\)\ (1\(\leq i\leq n).\) The condition A is very hard to check, while (*) has a meaning if \(\phi (s)>0\). The following equality also holds under the coordinate change (*): \(f(y_ 1,...,y_ n)=f(x_ 1,...,x_ n)+\sum^{\infty}_{j=1} (j!)^{-1}s^ j(f(x_ 1,..., x_ n)).\)\ I could not give a counter-example for the author's results themselves. Thus these misunderstandings and misprints may have been made when he wrote the final version of the manuscript. Perhaps he had forgotten the exact arguments which he had used at the earlier stage. compound Du Val singular points; canonical singularities; Newton diagrams Markushevich, DG, Canonical singularities of three-dimensional hypersurfaces, Math. USSR-Izv., 26, 315-345, (1986) Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, Singularities in algebraic geometry Canonical singularities of three-dimensional hypersurfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The authors study the nilpotent part of a pseudo-periodic automorphism \(h\) of a real oriented surface \(\Sigma\) with boundary \(\partial \Sigma\). Let \(e\) be the least common multiple of the orders of \(h\) restricted to each periodic piece. Consider the operator \(N\colon H_1(\Sigma,\partial \Sigma;\mathbb Z)\to H_1(\Sigma;\mathbb Z)\) defined by \(N([\gamma])=[h^e(\gamma)-\gamma]\). The quadratic form \(Q(v,w)=\langle Nv,w\rangle\) descends to a quadratic form \(\widetilde Q\) on \(H_1(\Sigma,\partial \Sigma;\mathbb Z)/\ker N\). A formula for \(Q\) is given, involving the so called screw numbers measuring the amount of rotation in the collection of annuli in the Nielsen-Thurston decomposition. The bilinear form \(\widetilde Q\) is positive definite if all the screw numbers associated to orbits of annuli whose core curves are non-nullhomotopic are positive. This condition is satisfied if \(h\) is the geometric monodromy of a reduced function \(f \colon (X, 0) \to (\mathbb C, 0) \) on a normal surface singularity. The screw numbers can then be computed in terms of the embedded resolution. Explicit computations and examples are given. Numerical invariants associated to \(\widetilde Q\) are able to distinguish the examples of pairs of plane curve singularities with different topological type but same spectral pairs, due to \textit{R. Schrauwen} et al. [Proc. Symp. Pure Math. 53, 305--328 (1991; Zbl 0749.14003)]. The form \(\widetilde Q\) is also computed for the two infinite families of reducible plane singularities that are not topologically equivalent but have the same Seifert form given by \textit{P. Du Bois} and \textit{F. Michel} [J. Algebr. Geom. 3, No. 1, 1--38 (1994; Zbl 0810.32005)]. In that case \(\widetilde Q\) is always defined on an abelian group of rank 4; while this form is a weaker invariant than the Seifert form it is also much easier to compute. pseudo-periodic surface automorphisms; positive definite quadratic form; monodromy automorphism; plane curve singularities; twist formula; screw numbers Singularities in algebraic geometry, Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Automorphisms of surfaces and higher-dimensional varieties On a quadratic form associated with a surface automorphism and its applications to singularity theory

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The multi-graded Hilbert scheme parametrizes all ideals with a given Hilbert function. An important subclass is the class of all toric Hilbert schemes. This paper details the main component of a toric Hilbert scheme in case it contains a point corresponding to an affine toric variety. Here the component can be viewed as the set of all flat limits of this variety. The main tool used in the paper is a computation of the fan of the toric variety. The treatment is very self-contained and thorough. Moreover, using a Chow morphism, a connection with certain GIT quotients of the toric variety is included. Some helpful examples are provided as well. toric Hilbert scheme; fiber polytope; toric Chow quotient Chuvashova O.V., The main component of the toric Hilbert scheme, Tôhoku Math. J., 2008, 60(3), 365--382 Parametrization (Chow and Hilbert schemes), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies The main component of the toric Hilbert scheme

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We develop, for finite groups, a certain kind of resolutions of quotient spaces \(\mathbb{C}^n/G\) called pluri-toric resolutions. The pluri-toric resolutions extend the use of toric geometry, from the study of quotients by abelian finite groups to the study of quotients by arbitrary finite groups. This is done in such a way that the combinatorical and stratifying nature of toric resolutions is inherited by the pluri-toric resolutions. We show that some of the properties of a pluri-toric resolution of \(\mathbb{C}^n/G\) can be deduced directly from the data of the group \(G\). One of these results states that a pluri-toric resolution, of a quotient by a finite subgroup of \(SL(n,\mathbb{C})\) where \(n\) is 2 or 3, is always crepant. The pluri-toric resolutions also give a generalised degree one McKay correspondence, i.e. a correspondence between certain conjugacy classes in \(G\) and the exceptional prime divisors in a pluri-toric resolution of \(\mathbb{C}^n/G\). The pluri-toric resolutions are constructed in two steps. In the first step we use a toric resolution of the quotient space \(\mathbb{C}^n/A\), where \(A\) is a maximal abelian subgroup of \(G\), to construct a partial resolution, called a mono-toric partial resolution, of \(\mathbb{C}^n/G\). In the second step we take a mono-toric partial resolution for each maximal abelian subgroup of \(G\) and patch these together to a pluri-toric resolution of \(\mathbb{C}^n/G\). Even if the construction addresses the general case we mainly focus on the cases when \(\mathbb{C}^n/G\) is a surface or a threefold. Our treatment leaves a number of open questions. resolutions of quotient spaces; pluri-toric resolutions; McKay correspondence Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Equisingularity (topological and analytic), Homogeneous spaces and generalizations Pluri-toric resolutions of quotient singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(S\) be a smooth projective surface defined over an algebraically closed field of positive characteristic and let \(H \in\text{Pic}(S)\). In this work the author studies the algebraic subset of \(|H|\) consisting of integral curves with singularities of prescribed types, called Severi variety. The main results give conditions on \(S\) and on the type and number of singularities for the Severi variety to be smooth and of the right dimension. The main tool used to obtain such theorems is a series of results by \textit{N. I. Shepherd-Barron} [Invent. Math. 106, No. 2, 243-262 (1991; Zbl 0769.14006)] about an extension to positive characteristic of Bogomolov's unstability theorem for rank 2 vector bundles on \(S\). singular curves; smooth surface; Severi variety; positive characteristic Singularities of curves, local rings, Local ground fields in algebraic geometry, Special surfaces Singular curves on smooth surfaces and Severi varieties in positive characteristic

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We construct a classification of coherent sheaves with an integrable log connection, or, more precisely, sheaves with an integrable connection on a smooth log analytic space \(X\) over \(\mathbb C\). We do this in three contexts: sheaves and connections which are equivariant with respect to a torus action, germs of holomorphic connections, and finally, global log analytic spaces. In each case, we construct an equivalence between the relevant category and a suitable combinatorial or topological category. In the equivariant case, the objects of the target category are graded modules endowed with a group action. We then show that every germ of a holomorphic connection has a canonical equivariant model. Global connections are classified by locally constant sheaves of modules over a (varying) sheaf of graded rings on the topological space \(X_{\log}\). Each of these equivalences is compatible with tensor product and cohomology. A.~Ogus, \emph{On the logarithmic {R}iemann-{H}ilbert correspondence}, Doc. Math. (2003), no.~Extra Vol., 655--724 (electronic), Kazuya Kato's fiftieth birthday. https://www.elibm.org/article/10011541 zbl 1100.14507 de Rham cohomology and algebraic geometry On the logarithmic Riemann-Hilbert correspondence

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities This paper concerns the study of families of smooth curves lying on a surface singularity \((S,O)\). Let \(\pi:X\to S\) be a minimal desingularization of \((S,O)\). The authors prove that for any irreducible component \(E\) of \(\pi^{-1}(0)\) such that \(\text{ord}_E (mO_X)=1\) (where \(\text{ord}_E\) is the divisorial valuation and \(m\) is the maximal ideal corresponding to \(O\) in \(S)\) the family of smooth curves \({\mathcal L}_E\) is nonempty. In that case a smooth curve in \({\mathcal L}_E\) corresponds to a point in \(E\). The authors conclude the existence of smooth curves lying on sandwich singularities. They also study wedge morphisms, that are morphisms \(h:\text{Spec} (k[[u,v]]) \to(S,O)\) such that the image of \(h\) is Zariski dense in some analytically irreducible component of \((S,O)\). arcs; resolution of singularities; surface singularity; desingularization; smooth curves lying on sandwich singularities; wedge morphisms Gonzalez-Sprinberg, Gérard; Lejeune-Jalabert, Monique, Families of smooth curves on surface singularities and wedges, Ann. Polon. Math., 67, 2, 179-190, (1997) Singularities of surfaces or higher-dimensional varieties, Plane and space curves, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Families of smooth curves on surface singularities and wedges

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let C be an irreducible plane algebroid curve singularity defined by \(f\in K[[x,y]]\), K algebraically closed. Let \(\Gamma\) be the value semigroup of C, n be an integer and \(A_ n=\{0=a_ 0<a_ 1<...<a_{n- 1}\}=\{\min \Gamma \cap \{k+n{\mathbb{Z}}_{\geq 0}\}| k=0,...,n-1\}\) the Apéry basis of \(\Gamma\) relative to n. Let \(h\in K[[x,y]]\) such that the intersection multiplicity \((f\cdot h)=\dim_ KK[[x,y]]/(f,h)\) is an element of \(A_ n\), \(n=ord_ xf(x,0)\). Then the author proves a factorization theorem for h with certain precise combinatorial statements about numerical invariants of the factors and numerical invariants of f. This theorem applies to h being a generic polar of f and some results of \textit{Merle} and \textit{Ephraim} turn out to be special cases of the factorization theorem of the author. Apéry basis of value semigroup; plane algebroid curve singularity Granja A.: Apéry basis and polar invariants of plane curve singularities. Pac. J. Math. 140(1), 85--96 (1989) Singularities of curves, local rings, Formal power series rings Apéry basis and polar invariants of plane curve singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(\mathcal{M}_{g}\) be the moduli space of Riemann surfaces of genus \(g\). Teichmüller curves can be obtained in \(\mathcal{M}_{g}\) for all \(g\) taking branched coverings of tori. A Teichmüller curve is said to be primitive if it does not arise from a curve in a moduli space of lower genus via a branched covering construction. Only a finite number of primitive Teichmüller curves have been found in each \(\mathcal{M}_{g}\). In this paper the author finds an infinite collection of primitive curves in genus two via the families of Jacobians they determine. Let \(X\) be a Riemann surface. A Weierstrass form is a holomorphic 1-form \(\omega \in \Omega (X)\) whose zero divisor is concentrated in a single point. In the case of genus two there are six such forms up to scale, one for each Weierstrass point. Let \(\mathcal{W}_{2}=\)\{\( X\in \mathcal{M}_{2}:\) Jac\((X)\) admits real multiplication with a Weierstrass eigenform\}. The author proves that the locus \(\mathcal{W}_{2}\) is a countable union of primitive Teichmüller curves. He exploits the connection between billiard tables and Teichmüller curves. We say \(P\) is a lattice polygon if SL\((X,\omega)\) is a lattice in SL\(_{2}(\mathbb{R})\) or, equivalently, if \((X,\omega)\) generates a Teichmüller curve. The author proves the following result on L-shaped polygons: a L-shaped polygon \( P(a,b)\) is a lattice polygon iff either \(a\) and \(b\) are rational or \(a=x+z\sqrt{d}\) and \(b=y+z\sqrt{d}\) for some \(x,y,z\in \mathbb{Q}\) with \(x+y=1\) and \(d\geq 0\) in \(\mathbb{Z}\). In the second case, the trace field of \(P(a,b)\) is \(\mathbb{ Q(}\sqrt{d})\). From this result it is proved that \(P(a,a)\) is a lattice polygon iff \(a\) is either rational or \(a=(1\pm \sqrt{d})/2\) for some \(d\in \mathbb{Q }\). As a consequence, every quadratic field arises as the trace field of a Teichmüller curve in \(\mathcal{M}_{2}\) and there are infinitely many primitive curves in \(\mathcal{M}_{2}\). The author also gives a direct algorithm to generate elements in SL\((X,\omega)\). In particular, he proves that the Teichmüller curve generated by \(P(a,a)\), \(a=(1\pm \sqrt{d})/2\) has genus zero for \(d=2,3,4,7,13,17,21,29\) and \(33\). Teichmüller curves; Hilbert modular surfaces; primitive curves in genus two; Weierstrass form McMullen, Curtis T., Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16, 4, 857-885, (2003) Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), , Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Modular and Shimura varieties Billiards and Teichmüller curves on Hilbert modular surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We discuss several invariants of complex normal surface singularities with a special emphasis on the comparison of analytic-topological pairs of invariants. Additionally we also list several open problems related with them. normal surface singularities; rational singularities; elliptic singularities; \( \mathbb{Q} \)-homology spheres; geometric genus; link of singularities; divisorial filtration; Poincaré series; Seiberg-Witten invariants of 3-manifolds; Casson invariant conjecture; Seiberg-Witten invariant conjecture; Heegaard-Floer homology; graded roots; surgery 3-manifolds; unicuspidal rational projective plane curves Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Applications of global analysis to structures on manifolds, Modifications; resolution of singularities (complex-analytic aspects) Pairs of invariants of surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities In J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009), \textit{M. Mustaţă} provides characterisations of KLT and LC pairs \((X, qY)\) with \(X\) smooth via dimension of jet schemes of \(Y\). The aim of the paper is to extend this result to the case \(X\) is normal and \(\mathbb{Q}\)-Gorenstein. KLT singularities; LC singularities Yasuda, T.: Dimensions of jet schemes of log singularities. Amer. J. Math. 125, No. 5, 1137-1145 (2003) Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Dimensions of jet schemes of log singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We define normalized versions of Berkovich spaces over a trivially valued field \( k\), obtained as quotients by the action of \( \mathbb{R}_{>0}\) defined by rescaling semivaluations. We associate such a normalized space to any special formal \( k\)-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed \( G\)-topological space, which we prove to be \( G\)-locally isomorphic to a Berkovich space over the field \( k((t))\) with a \( t\)-adic valuation. These spaces can be interpreted as non-archimedean models for the links of the singularities of \( k\)-varieties, and allow us to study the birational geometry of \( k\)-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field \( k\) is analogous to the structure of non-archimedean analytic curves over \( k((t))\) and deduce characterizations of the essential and of the log essential valuations, i.e., those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor. Rigid analytic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Normalized Berkovich spaces and surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(P_1,\dots,P_n \in \mathbb{C}^2\) be an ordered \(n\)-tuples of points in the plane. Assigning the points coordinates \(x_1,y_1,\dots,x_n,y_n\) the author identify the space \(E\) of all \(n\)-tuples \((P_1,\dots,P_n)\) with \(\mathbb{C}^{2n}\,.\) The locus \(V=\bigcup_{i<j} V_{ij}\) where two points coincide is a subspace arrangement in \(E\); its defining ideal \(I=I(V)=\bigcap _{i<j}(x_{i} - x_{j}, y_{i}-y_{j})\) is doubly homogeneous with respect to the double grading giving by degrees in the \({\mathbf x}=x_1,\ldots,x_n\) and \(\mathbf{y}=y_1,\dots,y_n\) variables separately. Moreover, the author shows that \(I^{m}=I^{(m)}\) and the Rees algebra \(R=\mathbb{C}[\mathbf{x},\mathbf{y}][tI]\) is Gorenstein. There are some open problems related to the study of rings of invariants and coinvariants for the action of the symmetric group \(S_n\) permuting the points among themselves. The author gives an exposition of some results by \textit{J.L. Martin} [Trans. Am. Math. Soc. 355, No.10, 4151--4169 (2003; Zbl 1029.05040)], related to the ideal of relations among the slopes of the lines that connect the \(n\) points pairwise. The appendix by E. Miller describes the Hilbert scheme \(H_n= \text{Hilb}^n(\mathbb{C}^2)\) of \(n\) points in the plane. Using results by Hartshorne, Haiman, Sturmfels, and Santos he shows that \(H_n\) is connected. The main result of this section says that \(H_n\) is a smooth and irreducible subvariety of the Grassmannian \(\text{Gr}^{n}(V_d)\) for \(d\geq n+1\). monomial ideal; Gorenstein ring; partition; complete graph Haiman, M.: Commutative algebra of n points in the plane. Trends Commut. Algebra, MSRI Publ. 51, 153--180 (2004) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, complete intersections and determinantal ideals, Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Commutative algebra of \(n\) points in the plane. Appendix by Ezra Miller: Hilbert schemes of points in the plane

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities [For the entire collection see Zbl 0644.00005.] L'A. présente dans ce texte les récents développements autour du théorème des zéros effectivs de Hilbert. Etant donnés des polynômes \(P_ 1,...,P_ m\in {\mathbb{C}}[X_ 1,...,X_ n]=:{\mathbb{C}}[X]\) de degrés \(\leq D\) et \(P\in {\mathbb{C}}[X]\) de degré \(D_ 0\) s'annulant sur l'ensemble des zéros communs de \(P_ 1,...,P_ m\) dans \({\mathbb{C}}^ n\), il s'agit de majorer un entier \(e\in {\mathbb{N}}\) et les degrés de polynômes \(A_ 1,...,A_ m\in {\mathbb{C}}[X]\) tels que \(P^ e=A_ 1P_ 1+...+A_ mP_ m\). L'A. a déjà établi de telles estimations du type \(<<_ n(D+D_ 0)^{\mu} \) où \(\mu =\min \{m,n\}\), contrastant avec les bornes précédentes de la forme \(<<_ n(D+D_ 0)^{2^{\mu}}\). Il explique dans ce texte comment obtenir la majoration plus fine \(e\leq (n+1)(\mu +1)(D_ 0+1)D^{\mu}\). Il pose également certaines questions dont la première a été récemment élucidée par \textit{C. Berenstein} et \textit{A. Yger} (``Effective Bézout identities in \({\mathbb{Q}}[X_ 1,...,X_ n]'')\), la seconde et la quatrième ont trouvé leurs réponses dans un travail de \textit{J. Kollár} [``Sharp effective Nullstellensatz'', J.Am. Math. Soc. 1, No.4, 963-975 (1988)] et une amélioration sensible en direction de la troisième a été obtenue dans le travail précité de Berenstein et Yger. Enfin, signalons qu'en direction opposée à l'exemple de Mayr et Meyer. \textit{F. Amoroso} a montré [``Tests d'appartenance'', C. R. Acad. Sci., Paris, Sér. I] que, sous des hypothèses raisonnables sur la variété des zéros de \(P_ 1,...,P_ m\) on peut effectivement écrire un polynôme \(Q\in (P_ 1,...,P_ m)\) sous la forme \(Q=A_ 1P_ 1+...+A_ mP_ m\) avec des bornes simples exponentielles pour les degrés des \(A_ i\). effective Hilbert zero theorem; effective Hilbert Nullstellensatz Relevant commutative algebra, Transcendence (general theory) Aspects of the Hilbert Nullstellensatz

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansion of two numbers \(\lambda >1\) and \(\lambda/(\lambda-1)\). continued fractions; surface singularities; Hirzebruch-Jung singularities; cups singularities; convex geometry; toric geometry; plumbing; JSJ decomposition Popescu-Pampu, P.: The geometry of continued fractions and the topology of surface singularities. Singularities in geometry and topology 2004, pp. 119--195, Adv. Stud. Pure Math. vol. 46, Math. Soc. Japan (2007) Singularities of surfaces or higher-dimensional varieties, Toric varieties, Newton polyhedra, Okounkov bodies, Complex surface and hypersurface singularities, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Topology of general 3-manifolds, Continued fractions The geometry of continued fractions and the topology of surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities \textit{F. Severi} claimed in the 1920s that the Hilbert scheme \({\mathcal H}_{d,g,r}\) of smooth irreducible non-degenerate curves \(C \subset \mathbb P^r\) of degree \(d\) and genus \(g\) is irreducible for \(d \geq g+r\) [Vorlesungen über algebraische Geometrie. Übersetzung von E. Löffler. Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)]. \textit{L. Ein} proved Severi's claim for \(r=3\) and \(r=4\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are counterexamples of various authors for \(r \geq 6\). It has been suggested by \textit{C. Ciliberto} and \textit{E. Sernesi} [in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 428--499 (1989; Zbl 0800.14002)] that Severi intended irreducibiity of the Hilbert scheme \({\mathcal H}^{\mathcal L}_{d,g,r} \subset {\mathcal H}_{d,g,r}\) of curves whose general member is linearly normal: indeed, the counterexamples above arise from families whose general member is not linearly normal. Here the authors prove irreducibility for \(g+r-2 \leq d \leq g+r\) (the Hilbert scheme is empty for \(d > g+r\) by Riemann-Roch) and for \(d=g+r-3\) under the additional assumption that \(g \geq 2r+3\). This extends work of \textit{C. Keem} and \textit{Y.-H. Kim} [Arch. Math. 113, No. 4, 373--384 (2019; Zbl 1423.14028)]. Hilbert scheme; linear series; linearly normal curves Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^r\) of relatively high degree

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(\chi\) be a numerical polynomial. Here we prove the connectedness of the real locus of the Hilbert scheme parametrizing all the subschemes of \(\mathbb{P}^r\) with \(\chi\) as Hilbert polynomial. connectedness of the real locus; Hilbert scheme Topology of real algebraic varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Connected and locally connected spaces (general aspects) The connectedness of the real locus of the Hilbert scheme of \(\mathbb{P}^ r\)

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(X_ g\) be the space of binary forms of degree \(g\), and let \(X_{p,g}\) be the subspace of forms having a root of multiplicity \(p\). \(X_ g\) can be identified with \(\text{Spec} R_ g\), where \(R_ g=\text{Sym} (S_ gV)\) for a fixed two-dimensional vector space \(V\) over \(\mathbb{C}\). Let \(J_{p,g}\) be the ideal of polynomials in \(R_ g\) vanishing on \(X_{p,g}\). For \(p=2\) it is well known that \(J_{p,g}\) is generated by one element of degree \(2g-2\), namely the discriminant. In this paper a formula for the dimensions of the graded pieces of \(J_{p,g}\) in the general case is derived. If \(g-1=(p-1) h+1\), \(0 \leq 1<p-1\), it is conjectured that \(J_{p,g}\) is generated by its elements of degrees \(2h\), \(2h+1\), and \(2h+2\). Hilbert functions of multiplicity ideals; binary forms Weyman, J.: On Hilbert functions of multiplicity ideals. J. algebra 161, 358-369 (1993) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Relevant commutative algebra, General ternary and quaternary quadratic forms; forms of more than two variables, General binary quadratic forms On the Hilbert functions of multiplicity ideals

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities A singularity is said to be exceptional, if for any log canonical boundary, there is at most one exceptional divisor of discrepancy \(-1\). This notion is important for the inductive treatment of the log canonical singularities. -- The exceptional singularities of dimension 2 are known: They belong to the types \(E_6\), \(E_7\), \(E_8\) after Brieskorn. In this paper it was proved that the quotient singularity defined by Klein's simple group in its 3-dimensional representations is exceptional. In the present paper, the classification of all the 3-dimensional exceptional quotient singularities is obtained. The main lemma states that the quotient of the affine 3-space by a finite group is exceptional if and only if the group has no semi-invariants of degree 3 or less. It is also proved that for any positive \(\varepsilon\), there are only finitely many \(\varepsilon\)-log terminal exceptional 3-dimensional quotient singularities. Theorem: A 2-dimensional quotient singularity \(X=\mathbb{C}^2/G\) by a finite group \(G\) without reflections is exceptional if and only if \(G\) has no semi-invariants of degree \(\leq 2\). Theorem: A 3-dimensional quotient singularity \(X=\mathbb{C}^3/G\) by a finite group \(G\) without reflections is exceptional if and only if \(G\) has no semi-invariants of degree \(\leq 3\). log canonical singularity; exceptional singularities; exceptional 3-dimensional quotient singularities Yu. G. Prokhorov and D. Markushevich, ''Exceptional Quotient Singularities,'' Am. J. Math. 121, 1179--1189 (1999). Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Homogeneous spaces and generalizations Exceptional quotient singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The authors compute the Euler characteristic of the Milnor fiber of an irreducible quasi-ordinary surface singularity \((X,O)\). More precisely, if \((X,O)\) is represented in a normalized coordinate system by \(f(x,y,z)=0\), then they show that the Euler characteristic of the Milnor fiber of \(f\) is the Euler characteristic of the plane curve singularity given by \(f(x,0,z)\), thus determined by its Puiseux pairs. In an added note they announce that they have a posteriori extended the results of the paper to calculate the zeta function of a reducible quasi-ordinary singularity of any dimension, applying the results of earlier papers: \textit{P. González Pérez}, ``Quasi-ordinary singularities via toric geometry'' (Thesis, Univ. La Laguna) and J. Inst. Math. Jussieu 2, No. 3, 383-399 (2003; Zbl 1036.32020). Euler characteristic; Milnor fiber; surface singularity Ban, C.; Mcewan, L. J.; Némethi, A.: On the Milnor fiber of a quasi-ordinary surface singularity, Canad. J. Math. 54, 55-70 (2002) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Milnor fibration; relations with knot theory, Topological properties in algebraic geometry On the Milnor fiber of a quasi-ordinary surface singularity

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities \textit{W. D. Neumann} and \textit{J. Wahl} proved in [Comment. Math. Helv. 65, No. 1, 58--78 (1990; Zbl 0704.57007) and Geom. Topol. 9, 757--811 (2005; Zbl 1087.32018)] that every surface singularity of the form \(f(x,y)=z^n\), whose link has the integral homology of a sphere, is analytically equivalent to a splice singularity. They also characterized such singularities among all singularities of the form \(f(x,y)=z^n\) in terms of their topological pairs. Since, by \textit{R. Mendris} and \textit{A. Némethi} [Compos. Math. 141, No.~2, 502--524 (2005; Zbl 1077.32016)], singularities of the form \(f(x,y)=z^n\) with rational homology sphere links are much more frequent, one could wonder if all or, at least, many of them are equivalent to splice quotient singularities. The answer is ``not at all'', as demonstrated in the present paper by characterizing all singularities of the form \(f(x,y)=z^n\), whose links are rational homology spheres, and whose topological types are those of splice quotient singularities. surface singularities; 3-dimensional homology spheres; link of a complete intersection surface singularity; rational homology spheres Sell, On the topology of surface singularities {zn = f(x, y)}, for f irreducible, Michigan Math. J. 59 (1) pp 85-- (2010) Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry, Complex surface and hypersurface singularities On the topology of surface singularities \(\{z^n=f(x,y)\}\), for \(f\) irreducible

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let $C$ be a smooth irreducible projective curve of genus $g$$\geq3$, $U_C^s(2,d)$ the moduli space of stable, degree $d$, rank $2$ vector bundles on $C$ and $B_d^k$ the Brill-Noether locus of vector bundles $\mathcal{F}$ of degree $d$ with $h^0(\mathcal{F})$$\geq{k}$. In the present paper the authors strenghten the result they proved in [\textit{Y. Choi} et al., Proc. Am. Math. Soc. 146, No. 8, 3233--3248 (2018; Zbl 1392.14003)]. where they showed that for $3$ $\leq$ $\nu$ $\leq$ $(g + 8 )/4$ and $3g - 1$ $\leq$ $d$ $\leq$ $4g - 6 -2$$\nu$ , $B^k_d$ $\cap$ $U^s_C(2,d)$ has exactly two reduced components $B_r$ and $B$$_s$ where $C$ is a general $\nu$-gonal curve. In this paper they improve the conditions on $\nu$ and $d$ and give more information on the components $B_r$ and $B_s$. Furthermore, they apply their main theorem to the study of Hilbert schemes of smooth surface scrolls in projective space. [\textit{C. Ciliberto} and \textit{F. Flamini}, Rev. Roum. Math. Pures Appl. 60, No. 3, 201--255 (2015; Zbl 1389.14013)], [\textit{M. Teixidor i Bigas}, Tohoku Math. J. (2) 43, No. 1, 123--126 (1991; Zbl 0702.14009)]. stable rank 2 bundles; Brill-Noether loci; general \(\nu \)-gonal curves; Hilbert schemes Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Rational and ruled surfaces, Parametrization (Chow and Hilbert schemes), Special divisors on curves (gonality, Brill-Noether theory), Special algebraic curves and curves of low genus Moduli spaces of bundles and Hilbert schemes of scrolls over \(\nu \)-gonal curves

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities [For part I see Trans. Am. Math. Soc. 309, No. 1, 309-323 (1988; Zbl 0682.14028).] The author studies the canonical ring of minimal algebraic surfaces with \(c^ 2_ 1=2p_ g-4\). These surfaces and their deformations were thoroughly studied by the reviewer [Ann. Math., II. Ser. 104, 357-387 (1976; Zbl 0339.14024)]; see also part I (loc. cit.) where the author studied the case \(c^ 2_ 1=2p_ g-3\). Reviewer's comments: Although it is not stated there, these canonical rings are already well understood in the paper cited above. The point is that most of these surfaces have a pencil \(| D|\) of curves of genus 2 (and the exceptional ones are easier). There is another curve \(G\) with \(DG=1\). One considers the bi-graded ring, the sum of \(R_{m,n}=H^ 0({\mathcal O}(mD+nG))\). This ring is easy to describe, and, since \(K\) is of the form \(rD+G\), the canonical ring is written down as its subring. In this manner, one need not to refer to any hyperplane section, and the result becomes more transparent compared to the one in the article. Horikawa surfaces Surfaces of general type On the canonical rings of some Horikawa surfaces. II

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities M. Reid gave a criterion saying which cyclic quotient singularities are terminal singularities. - The authors use a combinatorial lemma - due to G. K. White and reproved in this note - to interpret this criterion in certain cases and they obtain thereby a complete description of isolated terminal cyclic quotient singularities in dimension three and of isolated Gorenstein terminal cyclic quotient singularities in dimension four. canonical singularities; Bernoulli functions; Gorenstein singularities; cyclic quotient singularities; terminal singularities D. R. Morrison and G. Stevens, ''Terminal quotient singularities in dimensions three and four,''Proc. Amer. Math. Soc.,90, No. 1, 15--20 (1984). Singularities in algebraic geometry, Local complex singularities, Group actions on varieties or schemes (quotients), Fibonacci and Lucas numbers and polynomials and generalizations Terminal quotient singularities in dimensions three and four

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities A classical theorem in arithmetic geometry due to Serre-Tate (first proved by Néron-Ogg-Shafarevich in the curve case) states that an abelian variety has good reduction if and only if the action of the Inertia group on the first étale \(\ell\)-adic cohomology is trivial. One implication holding for all varieties alike, the converse is in general not true as can be seen already from curves of genus greater than one. There are, however, partial results, for instance for \(K3\) surfaces [\textit{C. Liedtke} and \textit{Y. Matsumoto}, Compos. Math. 154, No. 1, 1--35 (2018; Zbl 1386.14138); \textit{B. Chiarellotto} et al., Proc. Lond. Math. Soc. (3) 119, No. 2, 469--514 (2019; Zbl 1427.14075)] and for semistable surfaces in the realm of the weight-monodromy conjecture [\textit{M. Rapoport} and \textit{Th. Zink}, Invent. Math. 68, 21--101 (1982; Zbl 0498.14010)]. The present paper combines ideas from the previous two approaches to treat the following special situation: Let \(K\) be a complete field with respect to a discrete valuation such that the residue field \(k\) is algebraically closed of characteristric \(p>2\). Then there is a unique ramified quadratic extension \(L/K\); it corresponds to a unique quadratic character \(\psi\). Assume given a smooth surface \(X\) over \(K\) which possesses an integral model \(\mathcal X\) over the valuation ring \(R\) whose special fibre \(\mathcal X_k\) has at worst ordinary double points as singularities (the mildest singularity on a surface, given locally by \(xy=z^2\)). At a singular point \(P\in \mathcal X_k\), a formal neighbourhood in \(\mathcal X\) is isomorphic to \[ R[[x,y,z]/(z^2+xy+r) \] for some \(r\in \mathfrak p\), the maximal ideal of \(R\). Fix the \(\mathfrak p\)-adic order \(n_P\) of \(r\), i.e. \[ r\in \mathfrak p^{n_P}\setminus \mathfrak p^{n_P+1}. \] (There also is an intrinsic definition, so this is well-defined.) Let \(g\) be the number of singular points \(P\) such that \(n_P\) is odd: \[ g = \#\{P\in \mathcal X_k; 2\nmid n_P\}. \] Then the author proves the following result, assuming that \(p\nmid \ell\): Theorem.\(H^2(X_{\bar K}, \mathbb Q_\ell)^{I_K}\) has codimension \(g\) inside \(H^2(X_{\bar K}, \mathbb Q_\ell)\). Moreover, the quotient is \(\psi\)-isotypic. The proof proceeds by exhibiting a semi-stable model over \(K\), if all \(n_P\) are even, or over \(L\) in case some \(n_P\) is odd. This is achieved by successive explicit blow-ups (much in the spirit of [\textit{C. Liedtke} and \textit{Y. Matsumoto}, Compos. Math. 154, No. 1, 1--35 (2018; Zbl 1386.14138)], but less complicated since there are only ordinary double points). Then the author applies the Rapoport--Zink spectral sequence and appeals to known cases of the weight-monodromy conjecture due to \textit{M. Rapoport} and \textit{Th. Zink} [Invent. Math. 68, 21--101 (1982; Zbl 0498.14010)] in mixed characteristic and [\textit{T. Ito}, Am. J. Math. 127, No. 3, 647--658 (2005; Zbl 1082.14021)] in equal characteristic. algebraic surfaces; ordinary double point singularity; local fields; integral model; weight-monodromy conjecture Singularities of surfaces or higher-dimensional varieties, Étale and other Grothendieck topologies and (co)homologies Ramification in the cohomology of algebraic surfaces arising from ordinary double point singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We study the relation between the graded stable derived categories of 14 exceptional unimodal singularities and the derived categories of \(K3\) surfaces obtained as compactifications of the Milnor fibers. As a corollary, we obtain a basis of the numerical Grothendieck group similar to the one given by \textit{W. Ebeling} and \textit{D. Ploog} [Math. Ann. 347, No. 3, 689--702 (2010; Zbl 1193.13014)]. Masanori Kobayashi, Makiko Mase, and Kazushi Ueda, A note on exceptional unimodal singularities and K3 surfaces, Int. Math. Res. Not. IMRN 7 (2013), 1665 -- 1690. Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories A note on exceptional unimodal singularities and \(K3\) surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities This is a pseudo-historical survey about some aspects of lens spaces and their relations with cyclic quotient singularities. References are ordered by the year of publication. Their list is not exhaustive. 3-manifold; surface singularity; lens space Weber, C.: Lens spaces among 3-manifolds and quotient surface singularities. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM (2018) \textbf{(to appear)} Topology of general 3-manifolds, History of manifolds and cell complexes, History of mathematics in the 20th century, Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes, Singularities of surfaces or higher-dimensional varieties, Local complex singularities Lens spaces among 3-manifolds and quotient surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(S={\mathbb{C}}[X_ 1,...,X_ n]\). Combined with his earlier results elsewhere the author completes in this paper the classification of all finite subgroups G of GL(n,\({\mathbb{C}})\) for which \(S^ G\) is a complete intersection (C.I.). The classification depends upon the following results of the author and others: (1) One may assume that \(G\subseteq SL(n,{\mathbb{C}})\); (2) if \(S^ G\) is a C.I. then \(S^ G\) is generated as a \({\mathbb{C}}\)-algebra by 2n-1 elements; (3) if \(S^ G\) is a C.I. then G is generated by \(\{g\in G| rank(g-id)\leq 2\};\) (4) if \(S^ G\) is a C.I. then for every homogeneous linear \(v\in S\), \(S^ H\) is a C.I., where H is the isotropy group of v in G. action of subgroups of general linear group; invariant space as; complete intersection Nakajima, H, Quotient singularities which are complete intersections, Manuscr. Math., 48, 163-187, (1984) Group actions on varieties or schemes (quotients), Complete intersections, Linear algebraic groups over the reals, the complexes, the quaternions, Singularities in algebraic geometry, Geometric invariant theory Quotient singularities which are complete intersections

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(g \geq 2\), and \(\mathcal{M}_g\) be the moduli space of surfaces of genus \(g\). This space has the structure of an orbifold given by the Teichmüller space \(\mathbb{T}_g\), and the action of the mapping class group that produces a covering \(\mathbb{T}_g \rightarrow \mathcal{M}_g\). Then, \textit{H. E. Rauch} proved in [Bull. Amer. Math. Soc. 69, 390--394 (1962; Zbl. 0106.28801)] that every point in the branch locus \(\mathcal{B}_g\) is topologically singular for \(g > 3\), while \(\mathcal{B}_2\) and \(\mathcal{B}_3\) contain both topologically singular and non-singular points. The present paper gives a new proof of these facts by using the uniformization of Riemann surfaces and of their automorphism groups by means of Fuchsian groups. As a byproduct, the authors obtain a singular point of \(\mathcal{B}_2\) missing in the list of Rauch. They also study some equisymmetric families of surfaces, showing the singular and non-singular points in their branch loci. Riemann surface; moduli space; orbifold; Teichmüller space Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (analytic), Compact Riemann surfaces and uniformization, Teichmüller theory for Riemann surfaces Topologically singular points in the moduli space of Riemann surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(\Gamma\subset \text{PU}(2,1)\) be an arithmetic lattice. In the non-cocompact case, one can study the Baily-Borel compactification \(X=\overline{\Gamma\backslash \mathbf{B}}\) which might be of special type, and if \(\Gamma\) is neat, the divisor \(D=X\setminus (\Gamma\backslash \mathbf{B})\) is a disjoint sum of elliptic curves. The author tackles the classification of such (irregular) surfaces with non-positive Kodaira dimension, by geometric methods. Let's denote \(\mathcal{T}\) the class of compact complex surfaces of the type \(X=U\cup D\) where \(U\) has the complex unit ball as universal holomorphic covering, and \(D\) is as above. It is proven that if \(Y\) is relatively minimal, irregular, with non-positive Kodaira dimension, then \(Y\) is dominated by a surface in \(\mathcal{T}\) if and only if it is abelian (and actually isogenous to a product of an elliptic curve). Results from \textit{R.-P. Holzapfel} [``Complex hyperbolic surfaces of abelian type'', Serdica Math. J. 30, No. 2--3, 207--238 (2004; Zbl 1062.11035)] which started this classification, and \textit{G. Tian} and \textit{S. T. Yau} [``Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry'', Mathematical aspects of string theory, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1, 574--629 (1987; Zbl 0682.53064)] are used throughout the paper. The first step is to prove that no irregular surface in \(\mathcal{T}\) has non-negative Kodaira dimension; then one has to treat the zero-dimensional case. A useful review of surface theory recalls that the latter happens only when its minimal model is abelian or hyperelliptic. One proceeds in the two above steps by reductio ad absurdum, and Tian-Yau's theorem provides the sought contradiction. special surfaces; modular varieties; Shimura varieties; Picard modular surfaces 10.4310/MRL.2008.v15.n6.a9 Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Diophantine inequalities Irregular ball-quotient surfaces with non-positive Kodaira dimension

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \((X, x)\) be a normal Gorenstein surface singularity. The author computes and expresses precisely Watanabe's \(m\)-th plurigenus \(\delta_m(X, x)\) [see \textit{K. Watanabe}, Math. Ann. 250, 65-94 (1980; Zbl 0414.32005) and \textit{J. Wahl}, J. Am. Math. Soc. 3, No. 3, 625-637 (1990; Zbl 0743.14026)]. As a consequence, he shows that the series \(\{\delta_m(X, x)\}_{m \in \mathbb N}\) is determined by finitely many of \(\delta_m(X, x)\). For related results see \textit{S. D. Cutkosky} and \textit{V. Srinivas}, Ann. Math., II. Ser. 137, No. 3, 531-559 (1993; Zbl 0822.14006). Gorenstein surface singularity Okuma T. The plurigenera of Gorenstein surface singularities. Manuscripta Math, 1997, 94: 187--194 Singularities of surfaces or higher-dimensional varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The plurigenera of Gorenstein surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities A complex surface germ with isolated singularities in \((\mathbb{C}^3, 0)\) is called a super-isolated singularity if its strict transform under the blowing up with the center at the origin \(0\in\mathbb{C}^3\) is smooth. The author proves that the Jordan form of the complex monodromy associated with a super-isolated hypersurface singularity is completely determined by the characteristic and two Jordan polynomials of the monodromy. Furthermore, both Jordan polynomials depend upon invariants of singular points belonging to the tangent cone of a super-isolated hypersurface singularity. This implies that there exist super-isolated hypersurface singularities of different topological types having the same characteristic polynomial and links with isomorphic fundamental groups. As a result the author disproves a conjecture of \textit{S. S.-T. Yau} verified for semi-quasihomogeneous isolated hypersurface singularities [Contemp. Math. 101, 303-321 (1989; Zbl 0699.14001)]. This paper is a detailed version of an earlier note by the author [C. R. Acad. Sci., Paris Ser. I 312, No. 8, 601-604 (1991; Zbl 0723.32016)]. Jordan form; complex monodromy; super-isolated hypersurface singularity Artal Bartolo, E., Forme de Jordan de la monodromie des singularités superisolées de surfaces, Mem. Amer. Math. Soc., 109, 525, (1994), x+84 Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Mixed Hodge theory of singular varieties (complex-analytic aspects), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Milnor fibration; relations with knot theory Jordan form of the monodromy of superisolated singularities of surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities This monograph provides a comprehensive introduction to the theory of complex normal surface singularities, with a special emphasis on connections to low-dimensional topology. In this way, it unites the analytic approach with the more recent topological one, combining their tools and methods. In the first chapters, the book sets out the foundations of the theory of normal surface singularities. This includes a comprehensive presentation of the properties of the link (as an oriented 3-manifold) and of the invariants associated with a resolution, combined with the structure and special properties of the line bundles defined on a resolution. A recurring theme is the comparison of analytic and topological invariants. For example, the Poincaré series of the divisorial filtration is compared to a topological zeta function associated with the resolution graph, and the sheaf cohomologies of the line bundles are compared to the Seiberg-Witten invariants of the link. Equivariant Ehrhart theory is introduced to establish surgery-additivity formulae of these invariants, as well as for the regularization procedures of multivariable series. In addition to recent research, the book also provides expositions of more classical subjects such as the classification of plane and cuspidal curves, Milnor fibrations and smoothing invariants, the local divisor class group, and the Hilbert-Samuel function. It contains a large number of examples of key families of germs: rational, elliptic, weighted homogeneous, superisolated and splice-quotient. It provides concrete computations of the topological invariants of their links (Casson(-Walker) and Seiberg-Witten invariants, Turaev torsion) and of the analytic invariants (geometric genus, Hilbert function of the divisorial filtration, and the analytic semigroup associated with the resolution). The book culminates in a discussion of the topological and analytic lattice cohomologies (as categorifications of the Seiberg-Witten invariant and of the geometric genus respectively) and of the graded roots. Several open problems and conjectures are also formulated. Normal Surface Singularities provides researchers in algebraic and differential geometry, singularity theory, complex analysis, and low-dimensional topology with an invaluable reference on this rich topic, offering a unified presentation of the major results and approaches. normal surface singularities; links of singularities; plumbed 3-manifolds; weighted homogeneous singularities; superisolated singularities; splice quotient singularities; Newton non-degenerate singularities; geometric genus; Seiberg-Witten invariant; periodic constant; Ehrhart theory; Lattice cohomology; equivariant Poincaré series Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Singularities of surfaces or higher-dimensional varieties Normal surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The main results are: Theorem 1. Let \(C\) be an irreducible smooth complete curve of genus \(g\geq 5\) over \(\mathbb{C}\), \(\theta\subset JC\) be the theta-divisor on the Jacobian \(JC\) of the curve \(C\) and \(C-C\) be the surface in \(JC\) consisting of the points \(x-y\in JC\), for all \(x,y\in C\). Then, the following equality holds: \(C-C=\{a\in JC|\;a+\text{sing} \theta\subset\theta\}\), where \(\text{sing} \theta\) is the subset of singular points in \(\theta\). Theorem 2. Let \(\hat\theta=\{\zeta_{g-1}\in\text{Pic}^{g-1}(C)|\;h^ 0(\zeta_{g-1})\geq 1\}\subset\text{Pic}^{g-1}(C)\) be the canonical model of the theta-divisor and \(\xi':=K-\xi\in\text{Pic}^{g- 1}(C)\) for all \(\xi\in\text{Pic}^{g-1}(C)\), where \(K\) is the canonical class of \(C\). Then, if \(g\geq 5\) and the curve \(C\) is not trigonal: \(C- C=\bigcap_{\xi\in W^ 1_{g-1}}(\hat\theta_ \xi+\hat\theta_{\xi'})\), where \(W^ 1_{g-1}=\{\zeta_{g- 1}\in\text{Pic}^{g-1}(C)|\;h^ 0(\zeta_{g-1})\geq 2\}\), and \(\hat\theta_{-\xi}\) denotes the translate of \(\hat\theta\) by the divisor \((-\xi)\in\text{Pic}^{g-1}(C)\). If \(C\) is trigonal, then \(C- C=(W^ 0_ 3-g^ 1_ 3)\cup(g^ 1_ 3-W^ 0_ 3)\). Corollary. If \(g\geq 5\), then \(C-C={\displaystyle{\bigcap_{{D\in| 2\theta|\atop \mu_ 0(D)\geq 4}}}}D\), where \(| 2\theta|\) denotes the complete linear system of the divisor \(2\theta\) and \(\mu_ 0(D)\) denotes the multiplicity of \(D\) at zero. Thus, the author gives an exhausting answer to the hypothesis by \textit{B. van Geemen} and \textit{G. van der Geer} [Am. J. Math. 108, 615-641 (1986; Zbl 0612.14044)]: Let \(\Gamma_{00}\) be the vector space of sections from \(H^ 0({\mathcal O}_{JC}(2\theta))\), which vanish with multiplicity \(\geq 4\) at \(0\in JC\), and \(F=\{x\in JC|\;s(x)=0\) for all \(s\in\Gamma_{00}\}\), then \(F=\{x-y|\;x,y\in C\}\) in the case \(g\geq 5\) (the cases \(g=1,2,3\) and 4 are considered in the work immediately). -- The author gives exhausting answers also to the following questions: 1. (Mumford) Let \(D\) be the class of divisors of degree 0 on \(C\) such that for all divisors \(E\) of degree \(g-1\), for which \(| E|\) is a pencil then either \(| D+E|\neq\emptyset\) or \(|- D+E|\neq\emptyset\). Is \(D\equiv a-b\) for some \(a,b\in C\) then? 2. (The author) Let \(D\) be the class of divisors of degree 0 on \(C\) such that for all divisors \(E\) of degree \(g-1\), for which \(| E|\) is a pencil, \(| D+E|\neq\emptyset\). Is \(D\equiv a-b\) for some \(a,b\in C\) then? Jacobi varieties; 2nd order theta functions; Pic; canonical model of theta divisor Welters, G. : '' The surface C - C on Jacobi varieties and second order theta functions '', Acta Math. 157 (1986) 1-22. Theta functions and abelian varieties, Picard schemes, higher Jacobians, Jacobians, Prym varieties The surface \(C\)-\(C\) on Jacobi varieties and 2nd order theta functions

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities For quotient singularities, the irreducible components of the (reduced) base space of the versal deformation are in one to one correspondence with certain partial resolutions, called \(P\)-resolutions. In this note, the author determines all \(P\)-resolutions for quotient singularities, in fact, with a simple lemma, he reduces the general problem to the special case of cyclic quotient singularities which he studied in a previous paper [cf. Singularity theory and applications. Part I: Geometric aspects of singularities. Proc. Symp., Warwick 1988/89, Lect. Notes Math. 1462, 302-319 (1991; Zbl 0747.14002)]. \(P\)-resolutions; quotient singularities Stevens, J., Partial resolutions of quotient surface singularities, Manuscripta Math., 79, 1, 7-11, (1993) Global theory and resolution of singularities (algebro-geometric aspects), Homogeneous spaces and generalizations, Modifications; resolution of singularities (complex-analytic aspects) Partial resolutions of quotient singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let D be a quaternionic algebra over a real quadratic number field and \(S^ K\) be a Shimura surface associated to D. We prove Tate's conjecture on algebraic cycles in \(S^ K\), namely, the Galois-subspace of the \(\ell\)-adic cohomology of \(S^ K\) is spanned by algebraic cycles and has a dimension equal to the order of pole of the L-function of \(S^ K\). We use a relative trace formula of Jacquet-Lai [\textit{H. Jacquet} and the author, C. R. Acad. Sci., Paris, Sér. I 296, 959-963 (1983; Zbl 0533.10028)] to transfer the result of Harder-Langlands-Rapoport to our case, otherwise we follow their proof of the same conjecture for Hilbert- Blumenthal surfaces. Tate conjecture on algebraic cycles; quaternionic algebra; Shimura surface; Galois-subspace of the \(\ell \)-adic cohomology; L-function DOI: 10.1007/BF01168162 Cycles and subschemes, Transcendental methods, Hodge theory (algebro-geometric aspects), Special surfaces, Representation-theoretic methods; automorphic representations over local and global fields Algebraic cycles on compact Shimura surface

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The author shows how the multigraded Hilbert scheme construction of \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] can be used to construct a quasi-projective scheme which parametrize left homogeneous ideals in the Weyl algebra having fixed Hilbert function. Fixing an integral domain \(k\) of characteristic zero, the Weyl algebra \(W=k \langle x_1, \dots, x_n, \partial_1, \dots, \partial_n \rangle\) has a \(k\)-basis consisting of the set \(\mathcal B = \{ x^\alpha \partial^\beta | \alpha, \beta \in \mathbb N^n \}\). If \(A\) is an abelian group, then any \(A\)-grading \(\mathbb N^n \to A\) on the polynomial ring \(S = k[x_1,\dots,x_n]\) extends to an \(A\)-grading \(\mathbb N^{2n} \to A\) on \(W\) by \(\deg (x^\alpha \partial^\beta) = \deg \alpha - \deg \beta\), which induces a decomposition \(W = \bigoplus_{a \in A} W_a\). Given a Hilbert function \(h:A \to \mathbb N\), the corresponding Hilbert functor \(H^h_W\) takes a \(k\)-algebra \(R\) to the set of homogeneous ideals \(I \subset R \otimes_k W\) such that \((R \otimes _k W_a)/I_a\) is a locally free \(R\)-module of rank \(h(a)\) for each \(a \in A\). The main theorem says that \(H^h_W\) is representable by a quasi-projective scheme over \(k\). The strategy of the proof is similar to that of Haiman and Sturmfels [loc. cit.], but there are some new behaviors regarding monomials in the Weyl algebra \(W\) not seen in the polynomial ring \(S\). An obvious difference is that a product of monomials in \(W\) need not be a monomial. Another difference is that \(W\) has infinite antichains of monomial ideals, unlike the polynomial case: see work of \textit{D. MacLagan} [Proc. Am. Math. Soc. 129, 1609--1615 (2001; Zbl 0984.13013)]. Moreover, the natural extension of Gröbner basis theory for \(S\) to \(W\) does not work well, so the author considers the initial ideal of a left ideal in the associated graded algebra \(\text{gr} W\) and uses Gröbner basis theory for \(W\) developed by \textit{M. Saito} et al. [Gröbner deformations of hypergeometric differential equations. Berlin: Springer (2000; Zbl 0946.13021)]. The arguments are well presented along with examples showing the novel points. Hilbert schemes; Weyl algebras Parametrization (Chow and Hilbert schemes), Rings of differential operators (associative algebraic aspects) Multigraded Hilbert schemes parametrizing ideals in the Weyl algebra

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The author gives an introductory exposition of the theory of complex plane curve singularities, including relevant commutative algebra, Puiseux series, the Milnor number, resolution of singularities and the Gauß-Manin connection. conductor; Gauß-Manin connection; isolated singularity; milnor number; nullstellensatz; Puiseux series Singularities of curves, local rings, Plane and space curves, Milnor fibration; relations with knot theory Singularities of plane algebraic curves

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities \textit{W. T. Wu} [Shuzue Jinzhan 8, 395-401 (1965)] defined Chern characteristic classes of algebraic varieties with arbitrary singularities. These classes are algebraical equivalence classes and are computable. In 1987, W. T. Wu has proved that the Miyaoka-Yau inequality \(3c_2-c_1^2\geq 0\) is valid for the surfaces with singularities in \(\mathbb{C}\mathbb{P}_3\). In this paper, the author proves that the Miyaoka-Yau inequality of Chern characters is valid for certain kind of algebraic surfaces with singularities, e.g., ruled surfaces with singularities or surfaces in \(\mathbb{C}\mathbb{P}_n\) with \(n>3\) if the aggregate of the tangent planes of the surface is properly irreducible. algebraic surface with singularities; Chern characteristic classes; Miyaoka-Yau inequality Singularities of surfaces or higher-dimensional varieties, Characteristic classes and numbers in differential topology, Singularities in algebraic geometry The Chern characters of surface with singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(S=\text{spec} R\) where \(R\) is a Noetherian normal complete local ring containing an algebraically closed field isomorphic to the residue field of \(R\). Such an \(S\) is called a normal surface singularity, and it is called a rational surface singularity if furthermore there is a desingularization \(\pi:X\to S\) such that the stalk of \(R^1\pi_*{\mathcal O}_X\) at the closed point is zero. Let \(\pi:X\to S\) be any desingularization of a rational surface singularity \(S\). In the present paper, normal \(S\)-schemes \(Y\) factoring \(\pi\) are related to complete ideals on \(S\) and the semifactorization theory for the latter is used to get a characterization of \(S\)-isomorphisms between \(S\)-schemes \(Y\). isomorphisms between schemes; normal surface singularity; rational surface singularity; desingularization; complete ideals; semifactorization Cossart, V.; Piltant, O.; Reguera-López, A. J.: On isomorphisms of blowing-ups of complete ideals of a rational surface singularity. Manuscripta math. 98, No. 1, 65-73 (1999) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Rational and birational maps On isomorphisms of blowing-ups of complete ideals of a rational surface singularity

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities If \(m_i(x)\) is the list of all homogeneous monomials of degree \(d\) in \(x=(x_0,x_1,x_2,x_3)\) the set of all complex surfaces of degree \(d\) in \(\mathbb{C} P^3\) is parametrized by \[ Z=\{(x,\lambda)\in \mathbb{C} P^3\times\mathbb{C} P^{k-1}\mid\sum\lambda_im_i(x)= 0\}. \] If \(Y\) is the set of singular values of the projection \(\pi: Z\to\mathbb{C} P^{k-1}\) then \(\mathbb{C} P^{k-1}-Y\) is path connected. If \(Z(\lambda)=\pi^{-1}(\lambda)\), the set of non-singular complex surfaces of degree \(d\) in \(\mathbb{C} P^3\) which are defined over \(\mathbb{R}\) consists of the \(Z(\lambda)\) with \(\lambda\in\mathbb{R} P^{k-1}-Y\). Denote by \(\overline Z(\lambda)\) the quotient of \(Z(\lambda)\) by complex conjugation. In general \(\overline Z(\lambda)\) and \(\overline Z(\mu)\), for \(\lambda,\mu\in\mathbb{R} P^{k-1}-Y\), are not necessarily diffeomorphic since \(\mathbb{R} P^{k-1}-Y\) may not be path connected. However, a generic path from \(\lambda\) to \(\mu\) in \(\mathbb{R} P^{k-1}\) effects a surgery -- one of two types described in [\textit{G. Wilson}, Topology 17, 53-73 (1978; Zbl 0394.57001)] -- each time it intersects \(Y\). The author analyzes this situation to obtain results about the diffeomorphism and handlebody type of the \(\overline Z(\lambda)\)'s. homogeneous monomials; complex surfaces; surgery; diffeomorphism; handlebody S. Akbulut, ''On quotients of complex surfaces under complex conjugation'',J. Reine Angew. Math.,447, 83--90 (1994). Topology of Euclidean 4-space, 4-manifolds, Surfaces and higher-dimensional varieties On quotients of complex surfaces under complex conjugation

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(V/\Gamma\) be a symplectic quotient singularity, i.e. a quotient of a finite dimensional vector space \(V\) by a linear action of a finite subgroup \(\Gamma \subset \mathrm {Sp}(V)\). The article under review concerns \(\mathbb{Q}\)-factorial terminalizations of \(V/\Gamma\), i.e. projective, crepant, birational morphisms \(\rho : Y \rightarrow V/\Gamma\) such that \(Y\) has only \(\mathbb{Q}\)-factorial terminal singularities, and symplectic resolutions of \(V/\Gamma\), i.e. smooth \(\mathbb{Q}\)-factorial terminalizations. By results of Namikawa, a symplectic quotient singularity admits finitely many \(\mathbb{Q}\)-factorial terminalizations, and if one of them is smooth then all are smooth. The main result is a formula for the number of non-isomorphic \(\mathbb{Q}\)-factorial terminalizations of \(V/\Gamma\), expressed in terms of the Calogero-Moser deformation of \(V/\Gamma\) and the Namikawa Weyl group associated to \(V/\Gamma\). From this theorem the author derives a more explicit formula or obtains a number of symplectic resolutions for all groups \(\Gamma\) such that \(V/\Gamma\) is known to admit a symplectic resolution. These are: the infinite series of wreath products \(\mathcal{S}_n \wr G\), where \(G \subset \mathrm {SL}(2,\mathbb{C})\), acting on \(V = \mathbb{C}^{2n}\), a 4-dimensional representation \(G_4\) of the binary tetrahedral group and a 4-dimensional representation of \(Q_8 \times_{\mathbb{Z}_2} D_8\). In the case of \(\mathcal{S}_n \wr G\) the result is a formula involving the description of the Weyl group associated to \(G\) via the McKay correspondence, and for the remaining two cases the author obtains 2 and 81 symplectic resolutions respectively. symplectic resolutions; symplectic reflection algebras; Orlik-Solomon algebras Bellamy, G., Counting resolutions of symplectic quotient singularities, Compos. Math., 152, 1, 99-114, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Deformations of associative rings, Poisson algebras Counting resolutions of symplectic quotient singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(F\) be a graded module over an AS-regular algebra of dimension 3, generated in degree 1 with 3 quadratic relations. The author verifies that the Hilbert scheme, in the sense of \textit{M. Artin} and \textit{J. J. Zhang} [Algebr. Represent. Theory 4, No. 4, 305--394 (2001; Zbl 1030.14003)], parametrizing quotients of \(F\) with a fixed Hilbert polynomial, is projective. Hilbert scheme; quantum plane Algebraic moduli problems, moduli of vector bundles, Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry Hilbert schemes for quantum planes are projective

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The author gives an explicit formula for the number of Jordan blocks of size two for the eigenvalue 1 of the monodromy for surface singularities of the form \(f_d + f_{d+k} +\dots\), in terms of invariants from the graph for an embedded \(\mathbb Q\)-resolution of the tangent cone \(V (f_d)\) and the graph of the semistable reduction. monodromy; toric partial resolution; mixed Hodge structure Martín-Morales, J, 2-Jordan blocks for the eigenvalue \(\lambda =1\) of yomdin-Lê surface singularities, C. R. Math. Acad. Sci. Paris, 353, 161-165, (2015) Variation of Hodge structures (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Complex surface and hypersurface singularities 2-Jordan blocks for the eigenvalue \(\lambda = 1\) of Yomdin-Lê surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Die Arakelov-Geometrie ist eine arithmetische Schnittheorie für algebraische Varietäten, welche über den ganzen Zahlen definiert sind und durch Hinzufügen entsprechender analytischer Daten ``kompaktifiziert'' werden. Eine zentrale Rolle in der Arakelov-Geometrie spielen die arithmetischen Selbstschnittzahlen von hermiteschen Geradenbündeln. Die arithmetische Selbstschnittzahl von logarithmisch singulären, hermiteschen Geradenbündeln ist in der bisher entwickelten arithmetischen Schnitttheorie nicht definiert. Eine solche verallgemeinerte arithmetische Selbstschnittzahl ergibt sich erst aus einer entsprechenden Erweiterung der arithmetischen Schnitttheorie. In der vorliegenden Arbeit wird eine solche Erweiterung der arithmetischen Schnitttheorie im Falle der Modulkurven und dem Modulformenbündel, versehen mit der Petersson-Metrik, durchgeführt und die entsprechende verallgemeinerte arithmetische Selbstschnittzahl berechnet. Arakelov theory; modular curves; modular surfaces; self intersection -, Über die arithmetischen Selbstschnittzahlen zu Modulkurven und Hilbertschen Modulflächen , Dissertation, Humboldt-Universität zu Berlin, Berlin, 1999. Arithmetic varieties and schemes; Arakelov theory; heights, Arithmetic aspects of modular and Shimura varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Modular and Shimura varieties On the arithmetical self intersection number for modular curves and Hilbert modular surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The paper deals with the classification up to equivariant isomorphy of complex-algebraic normal affine surfaces endowed with an algebraic action of the multiplicative group \(\mathbb{C}^*\) of nonzero complex numbers. There are three different types of such surfaces, which are investigated separately, elliptic, parabolic and hyperbolic surfaces. In every case there is a quotient projection (in the elliptic case outside the unique fixed point) onto a smooth curve, which generically is either a \(\mathbb{C}^*\)-principle bundle (in the elliptic and hyperbolic case) or a line bundle (in the parabolic case). The main point of interest is thus the description of the exceptional fibres of this projection and their invariant neighbourhoods; the main tool here is Luna's slice theorem. Furthermore, for a given quotient curve every distribution of exceptional fibres is realized by an affine \(\mathbb{C}^*\)-surface. Finally the paper closes with a discussion of quotient singularities in the context of \(\mathbb{C}^*\)-surfaces. classification up to equivariant isomorphy; elliptic surface; parabolic surface; hyperbolic surfaces; exceptional fibres; quotient singularities Karl-Heinz Fieseler and Ludger Kaup, On the geometry of affine algebraic \?*-surfaces, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 111 -- 140. Families, moduli, classification: algebraic theory, Group actions on varieties or schemes (quotients), Transcendental methods of algebraic geometry (complex-analytic aspects) On the geometry of affine algebraic \(\mathbb{C}^*\)-surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(k\) be an algebraically closed field. The \(T_{44}\)-singularity is defined by \(R:=k[[X,Y]]/\langle F\rangle\), \(F=XY(X-Y)(X-\lambda Y)\), \(\lambda \in k \setminus \{0,1\}\). In their article [Arch. Math. 108, No. 6, 569--579 (2017; Zbl 1370.13011)], the authors describe for a wide class of Cohen-Macaulay \(R\)-modules the corresponding matrix factorizations. In the present article the description is accomplished. All together this gives a classification of all Cohen-Macaulay \(R\)-modules via matrix factorizations. Cohen-Macaulay modules; matrix factorizations; curve singularities Cohen-Macaulay modules, Singularities of curves, local rings Cohen-Macaulay modules over the plane curve singularity of type \(T_{44}\). II

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities A natural question arising in the study of algebraic curves is whether the Hilbert scheme of locally Cohen-Macaulay projective space curves of given degree and arithmetic genus is connected. The answer is unknown at present. In the paper under review, the author gives a nice survey of the current state of this question. projective space curve; degree; extremal curves; liaison; Rao module; triad Hartshorne, R.: Questions of connectedness of the Hilbert scheme of curves in \$\$\{\(\backslash\)mathbb\{P\}\^3\}\$\$ . In: Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), pp. 487--495. Springer, Berlin (2004) Parametrization (Chow and Hilbert schemes), Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Questions of connectedness of the Hilbert scheme of curves in \(\mathbb P^3\)

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The classical Hilbert scheme introduced by \textit{A. Grothendieck} [Sém. Bourbaki 1960/1961, Exp. 221 (1961; Zbl 0236.14003)], parametrizes subschemes of \(\mathbb P^r_k, k\) a field, with a given Hilbert polynomial. A toric variety is a variety parametrized by a finite number of monomials \({\mathbf t}^{a_i} = t_1^{a_1}\cdots t_d^{a_d}, i = 1,\ldots,n,\) in the polynomial ring \(k[t_1,\ldots,t_d],\) where \({\mathcal A} = \{a_1,\ldots,a_n\}\) denotes a subset of \(\mathbb N^d \setminus 0\) of \(n\) different vectors. Let \(S = k[x_1,\ldots, x_n]\) denote the polynomial ring in the variables \(x_1,\ldots,x_n\) of degree \(a_1,\ldots, a_n\) respectively. The toric ideal \(I_{\mathcal A}\) is the kernel of the natural map \(S \to k[t_1,\ldots, t_d], x_i \mapsto {\mathbf t}^{a_i}, i = 1,\ldots, n,\) which is a prime \(\mathbb N^d\)-graded ideal. A homogeneous ideal \(M \subset S\) is called \(\mathcal A\)-graded if \(\dim_k (S/M)_b = 1\) if \(b \in \mathbb N \mathcal A\) and \(0\) otherwise. That is, \(S/M\) has the same multigraded Hilbert function as the toric ring \(S/I_{\mathcal A}.\) The authors construct the toric Hilbert scheme \(H_{\mathcal A}\) that parametrizes all ideals with the same multigraded Hilbert function as \(S/I_{\mathcal A},\) satisfying a universality property. It follows that there exists exactly one component containing the point \([I_{\mathcal A}].\) If char\((k)= 0,\) then this component is reduced and so the point \([I_{\mathcal A}]\) on \(H_{\mathcal A}\) is smooth. Moreover, in the case of codim\((S/I_{\mathcal A}) = 2\) the authors prove the following additional results: (1) The toric Hilbert scheme has one component. It is the closure of the orbit of the toric ideal under the torus action. (2) The toric Hilbert scheme is 2-dimensional and smooth. Note that there is no restriction on the characteristic of the field \(k.\) (3) \(H_{\mathcal A}\) is the toric variety of the Gröbner fan of \(I_{\mathcal A}.\) In an unpublished paper, \textit{B. Sturmfels} started with a different construction in order to parametrize all ideals with the same Hilbert function as \(I_{\mathcal A}\) [``The geometry of \(\mathcal A\)-graded algebras'', preprint, \texttt{http://arxiv.org/abs/math.AG/94100032}] . toric variety; multigrading Peeva I., Stillman M., Toric Hilbert schemes, Duke Math. J., 2002, 111(3), 419--449 Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Toric varieties, Newton polyhedra, Okounkov bodies Toric Hilbert schemes.

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The author answers, in the case of rational surfaces with a conic bundle structure \(X/{\mathbb{P}}^ 1_ k\), several statements conjectured by \textit{J.-L. Colliot-Thélène} and \textit{J.-J. Sansuc} [cf. Duke Math. J. 48, 421-447 (1981 Zbl 0479.14006)]. More precisely let X be a smooth projective, geometrically integral surface over a perfect \(field\quad k,\) and \(\bar F\) the function field of \(\bar X=\bar k\times_ kX\). A map from \(K_ 2(\bar F)\) to the sum of function fields of all irreducible divisors on X is constructed. Let M be the cokernel of this map. - In this paper it is shown that in the case X is a rational surface with conic bundle structure \(X/{\mathbb{P}}^ 1_ k\) the kernel of the natural map \(H^ 1(Gal(\bar k/k),M)\) to \(\prod_{v}H^ 1(Gal(\bar k_ v/k_ v),M_ v) \) is 0 (where \(k_ v\) is the completion at a place v of k). - As a corollary it is shown that if furthermore \(H^ 1(Gal(\bar k,k),Pic(\bar X))=0\) then there exists a 0-cycle of degree \( 1\) on X if and only if there are 0-cycles of degree \( 1\) on \(X_ v\) at all places v of k. It is also explained how to calculate the order of \(A_ 0(X)\). zero-cycles; K-theory; Chow group; rational surface with conic bundle structure P. Salberger, ''Zero-cycles on rational surfaces over number fields,'' Invent. math., vol. 91, iss. 3, pp. 505-524, 1988. Algebraic cycles, Rational and unirational varieties, Applications of methods of algebraic \(K\)-theory in algebraic geometry Zero-cycles on rational surfaces over number fields

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities A three-dimensional orbifold \((\Sigma, \gamma_i, n_i)\), where \(\Sigma\) is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair \((X, C)\), where \(X\) is a normal surface singularity with QHS link and \(C\) is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, proving the existence of the universal abelian log cover of a pair. A key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with the case where \(C\) is empty and one considers the universal abelian cover, under certain conditions on a resolution graph one can construct pairs and their universal abelian log covers. Such pairs are called orbifold splice quotients. surface singularity; splice quotient singularity; orbifold homology; rational homology sphere; singular pair; abelian cover Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities Orbifold splice quotients and log covers of surface pairs

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Authors' abstract: We prove a closed formula for leading Gopakumar-Vafa BPS invariants of local Calabi-Yau geometries given by the canonical line bundles of toric Fano surfaces. It shares some similar features with Göttsche-Yau-Zaslow formula: Connection with Hilbert schemes, connection with quasimodular forms, and quadratic property after suitable transformation. In Part I of this paper we will present the case of projective plane, more general cases will be presented in Part II. Gopakumar-Vafa BPS invariant; Gromov-Witten invariants; local Calabi-Yau; Hilbert scheme; quasimodular form S. Guo, J. Zhou, Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. II, preprint. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects) Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. I

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(S\) be a smooth complex projective surface of general type and let \(f\) be its canonical map, i.e. the map associated with the canonical system \(|K_S |\) of \(S\). Assume that \(\dim f(S) = 1\) and let \(S @>g>>B \to f(S)\) be the Stein factorization of \(f\). In view of Beauville's result \(g\) is a fibration whose general fibre \(F\) has genus \(g(F) = 2,3,4,5\), under some assumption. The paper under review is a contribution to the study of \(g\) especially when \(g(f) = 4,5\) and to the problem of finding the largest \(c(g(F))\) such that \(K^2_S \geq c(g(F)) \cdot h^0 (K_S) +\) constant. genus of fibre; surface of general type; canonical map Sun X.T.: On canonical fibrations of algebraic surfaces. Manuscr. Math. 83, 161--169 (1994) Surfaces of general type, Structure of families (Picard-Lefschetz, monodromy, etc.) On canonical fibrations of algebraic surfaces

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let L be a very ample divisor on a smooth projective surface S, and let K denote the canonical divisor on S. Sommese and Van de Ven have proved that the linear system \(| L+K|\) is base-point free unless \((S,L)=({\mathbb{P}}^ 2,{\mathcal O}(i))\), \(i=1\) or 2, or \((S,L)\) is a scroll. The morphism \(\phi_{L+K}\) defined by \(| L+K|\) is called the adjunction mapping (associated to L). Let \(S\to^{\alpha}\hat S\to^{\beta}{\mathbb{P}}^ n\) be the Stein factorization of \(\phi_{L+K}\). The author concentrates on the structure of the morphism \(\beta\) and proves the following results: Theorem 1: Suppose that \(\dim(\phi_{L+K}(S))=2\) and let K and S as above. Then \(\beta\) is an embedding except in the following cases: (i) S is obtained by blowing-up \({\mathbb{P}}^ 2\) at 7 points in general position, and either \(S=\hat S\) and \(L=-2K\), or the map \(\alpha: S\to \hat S\) is the blowing-up of \(\hat S\) at one point P and \(L=\alpha^*(-2\hat K)-E\), where \(E=\alpha^{-1}(P)\); (ii) S is obtained by blowing-up \({\mathbb{P}}^ 2\) at 8 points in general position, and \(L=-3K\); (iii) \(S={\mathbb{P}}({\mathcal E})\) is the projectivization of a rank 2 vector bundle \({\mathcal E}\) over an elliptic curve Y, where \({\mathcal E}\) is a non-split extension \(0\to {\mathcal O}_ Y\to {\mathcal E}\to {\mathcal O}_ Y(P)\to 0,\) \(P\in Y\). If \(\zeta ={\mathcal O}_{{\mathbb{P}}({\mathcal E})}(1)\) is the tautological invertible sheaf, then L is numerically equivalent to \(3\zeta\). - One observes that the finite map \(\beta\) is 2-to-1 in case (i), and 3-to-1 in cases (ii) and (iii). Theorem 2: There exists an effective divisor \(C\in | L|\) which is a smooth hyperelliptic curve if and only if (S,L) belongs to one of the following cases: (a) \(({\mathbb{P}}^ 2,{\mathcal O}(i))\) with \(i=1, 2\) or 3; (b) S is a geometrically ruled surface over a hyperelliptic curve, and the restriction of L to a fibre has degree 2; (c) S is a rational ruled surface, and the restriction of L to a fibre has degree 2; (d) \(S={\mathbb{P}}({\mathcal E})\) is the projectivization of a rank 2 vector bundle \({\mathcal E}\) over an elliptic curve Y, where \({\mathcal E}\) is a non-split extension \(0\to {\mathcal O}_ Y\to {\mathcal E}\to {\mathcal O}_ Y(P)\to 0,\) with \(P\in Y\). - If \(\zeta ={\mathcal O}_{{\mathbb{P}}({\mathcal E})}(1)\) denotes the tautological invertible sheaf, and F is a fibre of \(S\to Y\), then L is numerically equivalent to \(2\zeta +F\); (e) \((S,L)\) is as described in cases (i) and (ii) of theorem 1. In cases (a),(b),(c) and (d), every smooth divisor \(D\in | L|\) is hyperelliptic, but in case (e) the general element of \(| L|\) is not hyperelliptic. In this paper some bounds are also given for the degree of the fibres of a ruled surface in \({\mathbb{P}}^ n\). It is proved that a hyperelliptic curve C of genus \(g>0\) can be embedded in the rational surface \({\mathbb{P}}({\mathcal O}_{{\mathbb{P}}^ 1}\oplus {\mathcal O}_{{\mathbb{P}}^ 1}(-e))\) if and only if \(e\leq g+1\). If C is a general hyperelliptic curve of genus g and \(e\leq g+1\), then the curves in \({\mathbb{P}}({\mathcal O}_{P^ 1}\oplus {\mathcal O}_{P^ 1}(-e))\) isomorphic to C move in an algebraic family of dimension \(g+6\). hyperelliptic divisors; very ample divisor; canonical divisor; adjunction mapping; degree of the fibres of a ruled surface SERRANO F., ''The adjunction mapping and hyperelliptic divisors on a surface'', J. Reine Angew. Math. 381 (1987), 90--109. Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Families, moduli, classification: algebraic theory The adjunction mapping and hyperelliptic divisors on a surface

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \((R,m)\) be a Gorenstein local ring, analytically unramified, \(I \subset R\) an \(m\)-primary ideal of \(R\), \(\overline I\) the integral closure of \(I\) and \(\delta (I)\) the reduction number of \(I\). There are some integers \(\{\overline e_d (I)\}\) such that \[ \text{length} (R/ \overline {I^{n + 1}}) = \overline e_0 (I) {n + d \choose d} - \overline e_1 (I) {n + d - 1 \choose d - 1} + \cdots + (- 1)^d \overline e_d (I) \] for sufficiently large \(n\). The paper gives sufficient conditions in terms of these invariants under which \(G(I) = R/I \oplus I/I^2 \oplus \ldots\) is Cohen-Macaulay or Gorenstein. -- Suppose \(\dim R = 1\), \(I\) is integrally closed and \(\overline e_1 (m) + \text{length} (R/ \overline m) = 1 + e(R)\). If \(e(R) = 2\) then \(G(I)\) is Gorenstein if and only if either \(I = m\), or \(I\) is the conductor \(C = (R : \overline R)\) of \(R\). If \(e(R) \leq 3\) then \(G(I)\) is Gorenstein if and only if either \(I = m\), or \(I = C\), or \(\delta (I) \leq 1\) and \(\text{length} (R/I) = e(R) - 1\). -- Now suppose \(\dim R = 2\). Then \(p_g (R) = \sup \{\overline e_2 (I) \mid I\) is an \(m\)-primary ideal\} is the geometric genus of \(R\). If \(R\) is pseudo-rational, that is \(p_g (R) = 0\), then the following conditions are equivalent: (i) \(G(I)\) is Gorenstein, (ii) \(\delta (I) \leq 1\) and \(e(I) = 2\cdot {length} (R/I)\). (iii) \(I\) is integrally closed and \(e(I) = 2\cdot {length} (R/I)\). If \(R\) is elliptic, that is \(p_g (R) = 1\) and \(I\) is integrally closed, then \(G(I)\) is Gorenstein if and only if either \(I = m\), or \(\delta (I) \leq 1\) and \(e(I) = 2\cdot {length} (R/I)\). -- If \(p_g (R) = 2\) then either \(e(R) = 2\), or \(\text{emb} (R) = e(R)\) and \(G(m)\) is Gorenstein. -- If \(p_g (R) = 3\) then \(G(m)\) is Gorenstein if and only if \(R\) is a quadratic hypersurface, a quartic hypersurface, or \(\text{emb} (R) = e(R)\). multiplicity; Cohen-Macaulay property; Gorenstein property; associated graded ring; elliptic curve; Gorenstein local ring; reduction number; quadratic hypersurface; quartic hypersurface Ooishi, A.: Tangent cones at curve and surface singularities. J. pure appl. Algebra 95, 189-201 (1994) Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Special algebraic curves and curves of low genus, Singularities of surfaces or higher-dimensional varieties Tangent cones at curve and surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Es handelt sich um die Fortsetzung der Arbeit: ``Der kombinatorische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' [\textit{G. Gotzmann}, Münster 1994; Zbl 0834.14004)]. Im vorliegenden zweiten Teil ist wieder \(k=\mathbb{C}\) der Grundkörper, \(\mathbb{H}=H_{d,g}\) das Hilbertschema der Kurven vom Grad \(d>1\) und Geschlecht \(g\) in \(\mathbb{P}^3_k\) und \(A_1(\mathbb{H})\) die erste Chowgruppe mit Koeffizienten in \(\mathbb{Q}\). Die Ergebnisse sind folgende, wobei zur Abkürzung \(g(d)=(d^2- 5d+10)/6\) gesetzt wird. Satz I. (i) Wenn \(d\geq 2\) und \(g\neq(d-1) (d-2)/2\) ist, dann ist \(\text{Pic}(\mathbb{H}) \simeq\mathbb{Z}^\rho \oplus \mathbb{C}^r\) mit \(\rho\geq 3\) und \(r=\dim_\mathbb{C} H^1(\mathbb{H}, {\mathcal O}_\mathbb{H})\). (ii) Wenn \(d\geq 8\) und \(g\leq g(d)\) ist, dann ist \(\rho\leq 4\). Satz II. (i) Wenn \(d\geq 2\) und \(g\neq(d-1) (d-2)/2\) ist, dann ist \(\dim A_1(\mathbb{H})\geq 3\). (ii) Wenn \(d\geq 8\) und \(g\leq g(d)\) ist, dann wird \(A_1(\mathbb{H})\) von bestimmten 1-Zykeln erzeugt. Chow group; Hilbert scheme; space curve Parametrization (Chow and Hilbert schemes), Plane and space curves The algebraic part of the first Chow group of a Hilbert scheme of space curves

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We study the structure constants of the class algebra \(R_{\mathbb Z}(\Gamma_n)\) of the wreath products \(\Gamma_n\) associated to an arbitrary finite group \(\Gamma\) with respect to the basis of conjugacy classes. We show that a suitable filtration on \(R_{\mathbb Z}(\Gamma_n)\) gives rise to the graded ring \({\mathcal G}_\Gamma(n)\) with non-negative integer structure constants independent of \(n\) (some of which are computed), which are then encoded in a Farahat-Higman ring \({\mathcal G}_\Gamma\). The real conjugacy classes of \(\Gamma\) come to play a distinguished role and are treated in detail in the case when \(\Gamma\) is a subgroup of \(\text{SL}_2({\mathbb C})\). The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface \(X\). wreath products; class algebras; Hilbert schemes Wang, W.: The Farahat-Higman ring of wreath products and Hilbert schemes. Adv. Math. 187, 417--446 (2004) Frobenius induction, Burnside and representation rings, Parametrization (Chow and Hilbert schemes), Classical real and complex (co)homology in algebraic geometry, Extensions, wreath products, and other compositions of groups The Farahat-Higman ring of wreath products and Hilbert schemes

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \((X,x)\) be a normal isolated singularity of dimension two over the complex number field \(\mathbb{C}\). The second pluri-genus \(\delta_2 (X,x)\) are studied in this article. Let \(m\) be a positive integer. Following Kimio Watanabe, the \(m\)-th pluri-genus \(\delta_m (X,x)\) is defined to be the integer \(\delta_m(X,x) =\dim_\mathbb{C} H^0({\mathcal O}_U (mK_U))/L^{2/m}(U)\), where \(X\) is an appropriate representative of the germ \((X,x)\), \(U=X-\{x\}\), \(K_U\) denotes the canonical divisor of \(U\), and \(L^{2/m}(U)\) denotes the \(\mathbb{C}\)-vector subspace of \(L^{2/m}\)-integrable \(m\)-ple holomorphic 2-forms on \(U\). \(\delta_1 (X,x)\) coincides with the geometric genus of \((X,x)\). Relations among the second pluri-genus, other various invariants and various conditions which the singularity satisfies are studied. As invariants, the geometric genus, the Milnor number, the Tjurina number, the modality, the inner modality and so on are considered. Such conditions as Gorenstein, Du Bois, hypersurface, complete intersection, quasi-homogeneous and so forth are treated. The following is one of Okuma's results: If \((X,x)\) is a Gorenstein singularity with \(p_g(X,x)\geq 1\), then \(\delta_2(X,x)=p_g(X,x)-(2K+E)\cdot(K+E)/2\), where \(p_g(X,x)\) denotes the geometric genus, \(K\) denotes the canonical divisor on the minimal good resolution of \((X,x)\), and \(E\) denotes the exceptional reduced divisor on the minimal good resolution of \((X,x)\). surface singularity; deformation; modality; pluri-genus; Milnor number; Tjurina number Okuma T. The second pluri-genus of surface singularities. Compos Math, 1998, 110: 263--276 Invariants of analytic local rings, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Modifications; resolution of singularities (complex-analytic aspects) The second pluri-genus of surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities This is a very useful survey on the topology, geometry and algebraic geometry associated to a complex normal surface singularity. Work in progress of W. Neumann and the author on the possible geometry and equations associated to a given simple link is described in an attractive way, by comparing it with the much simpler case of curve singularities. complex normal surface singularity equations; topology; link; resolution J. Wahl, Topology, geometry, and equations of normal surface singularities, in: Singularities and Computer Algebra, London Mathematical Society Lecture Note Series Vol. 324 (Cambridge University Press, Cambridge, 2006), pp. 351-371. Singularities of surfaces or higher-dimensional varieties, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Topology of general 3-manifolds, Knots and links in the 3-sphere Topology, geometry, and equations of normal surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga-Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander-Solberg and Kong type results. Frobenius categories; Iwanaga-Gorenstein algebras; Gorenstein-projective modules; Cohen-Macaulay modules; non-commutative resolutions; singularity categories; rational surface singularities M.~Kalck, O.~Iyama, M.~Wemyss, and D.~Yang. Frobenius categories, {G}orenstein algebras and rational surface singularities. {\em Compos. Math.}, 151(3):502--534, 2015. DOI 10.1112/S0010437X14007647; zbl 1327.14172; MR 3320570; arxiv 1209.4215 Singularities of surfaces or higher-dimensional varieties, Cohen-Macaulay modules, Derived categories, triangulated categories, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) Frobenius categories, Gorenstein algebras and rational surface singularities

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities We study the intersection theory on surfaces with abelian quotient singularities and we obtain formulas for its behavior under weighted blow-ups. As applications, we extend Mumford's formulas for the intersection theory on normal divisors, we derive properties for quotients of weighted projective planes, and finally, we compute abstract \(\mathbf Q\)-resolutions of normal surfaces using Jung's method. quotient singularity; intersection number; embedded \(\mathbf Q\)-resolution Artal, E., Martín-Morales, J., Ortigas-Galindo, J.: Intersection theory on abelian-quotient \textit{V}\ -surfaces and Q-resolutions. J. Singul. 8, 11-30 (2014) Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry Intersection theory on abelian-quotient \(V\)-surfaces and \(\mathbf Q\)-resolutions

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities Let \(k\) be a complete nonarchimedean valued field with a nontrivial valuation and let \(k^\circ\) be its ring of integers. The paper under review contains finiteness results for the étale cohomology of compact \(k\)-analytic spaces (in the sense of Berkovich) and for vanishing cycles on formal schemes over \(k^\circ\). We now give precise statements. Let \(X\) be a compact \(k\)-analytic space and let \(F\) be an abelian constructible sheaf on \(X\) with torsion orders prime to the residue characteristic. Assume that \(k\) is algebraically closed. Then, the groups \(H^q(X,F)\) are finite for \(q\geq 0\). Let \(\mathfrak{X}\) be a formal scheme locally topologically of finite presentation over \(k^\circ\) and let \(F\) be an abelian constructible sheaf on \(\mathfrak{X}_{\eta}\) with torsion orders prime to the residue characteristic. Then the vanishing cycles sheaves \(R^q \Psi_{\eta}(F)\) are constructible for \(q\geq 0\). The last results concern a more general class of formal schemes, namely special formal schemes, i.e. locally of the form Spf\((A)\), where \(A\) is a quotient of an algebra of the form \(k^\circ\{T_{1},\dots,T_{n}\}[[S_{1},\dots,S_{m}]]\). In this setting, the above result still holds if \(k\) is assumed to be discretely valued and if the constructibility condition is replaced by the more restrictive notion of \(\mathfrak{X}\)-constructibility. Let us mention that similar results were proven by \textit{R. Huber} [J. Algebr. Geom. 7, No. 2, 313--357 (1998; Zbl 1040.14008); J. Algebr. Geom. 7, No. 2, 359--403 (1998; Zbl 1013.14007)] under the assumption that \(k\) has characteristic 0. Note that \textit{V. Berkovich} himself had already proven similar results under algebraicity assumptions in [Invent. Math. 115, No. 3, 539--571 (1994; Zbl 0791.14008); ibid. 125, No. 2, 367--390 (1996; Zbl 0852.14002)]. Actually the proofs in the present paper rely on those former results, the extra hypotheses being removed thanks to Gabber's weak uniformization (see [\textit{L. Illusie} (ed.) et al., Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l'École Polytechnique 2006--2008. Paris: Société Mathématique de France (SMF) (2014; Zbl 1297.14003)]) and Deligne's cohomological descent. Finally, as regards weak uniformization, let us mention that the author proves results of this kind in the settings he considers: \(k\)-analytic spaces (covered by generic fibers \(\hat{\mathcal{Y}}_{\eta}\) of schemes \(\mathcal{Y}\) over \(k^\circ\) with \(\mathcal{Y}_{\eta}\) smooth over \(k\)) and special formal schemes over \(k^\circ\) (covered by completions of semi-stable schemes over rings of integers of finite extensions of \(k\)). For this purpose, applying Gabber's results directly is sometimes not enough and the author needs to go back to the proof and adapt the arguments. Berkovich spaces; formal schemes; étale cohomology; vanishing cycles; weak uniformization 10.1007/s11856-015-1249-6 Rigid analytic geometry, Étale and other Grothendieck topologies and (co)homologies Finiteness theorems for vanishing cycles of formal schemes

Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities The author applies the theory of abelian covers [\textit{R. Pardini}, J. Reine Angew. Math. 417, 191--213 (1991; Zbl 0721.14009)] to construct some examples of surfaces of general type, the most interesting of which is a surface with \(p_g=4\), \(K^2=31\) and birational canonical map. The examples are constructed as desingularizations of singular abelian covers of \({\mathbb P}^1\times{\mathbb P}^1\). abelian cover; surface of general type; canonical map Liedtke C. (2003) Singular abelian covers of algebraic surfaces. Manuscr. Math. 112(3): 375--390 Surfaces of general type, Singularities of surfaces or higher-dimensional varieties, Special surfaces, Coverings in algebraic geometry Singular abelian covers of algebraic surfaces