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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 The canonical rings of certain minimal algebraic surfaces are studied. They are of general type with \(c^ 2_ 1=2p_ g-3\), and for which the canonical system \(| K|\) has one base point. That this is the case except for a few cases is proved by the reviewer [Invent. Math. 37, 121- 155 (1976; Zbl 0339.14025)].
In the paper under review, the canonical ring of such surface is described in terms of generators and relations. The author deduces the result from the structure of the canonical ring of a canonical curve on S, which is hyperelliptic in these cases. Hence the result apparently depends on the choice of the canonical curve.
It seems to the reviewer that the result may be obtained more directly from the work cited above, without appealing to canonical curves, so that the representation is in a more invariant form. Also, E. Griffin's thesis treats the same problem for numerical quintics, giving a new construction of deformations of such surfaces [\textit{E. E. Griffin} II, Compos. Math. 55, 33-62 (1985; Zbl 0578.14033)]. Horikawa surface; canonical rings; minimal algebraic surfaces Surfaces of general type, Divisors, linear systems, invertible sheaves On the canonical rings of some Horikawa surfaces. I | 0 |
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 complex algebraic surface imbedded in \(P^ N({\mathbb{C}})\) and denote by S the set of singular points of X. The author proves that the harmonic \(L^ 2\)-spaces on X-S have pure Hodge and hard Lefschetz structures, provided S consists of isolated points. The Kähler metric used to define the harmonic \(L^ 2\)-spaces is the one induced from the Fubini-Study metric on \(P^ N({\mathbb{C}})\). The pure Hodge structure is obtained for \(i=p+q\neq 2\) and is conjectured for \(i=p+q=2\). harmonic \(L^ 2\)-spaces; hard Lefschetz structures; pure Hodge structure Masayoshi Nagase, Pure Hodge structure of the harmonic \?²-forms on singular algebraic surfaces, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 1005 -- 1023 (1989). Transcendental methods, Hodge theory (algebro-geometric aspects), Ordered rings Pure Hodge structure of the harmonic \(L^ 2\)-forms on singular algebraic surfaces | 0 |
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 An usual way to study the complexity of a local noetherian ring \((A,m)\) is in terms of its associated category \(CM(A)\) of Maximal Cohen-Macaulay (MCM) finitely generated modules (recall that an \(A\)-module \(M\) is Maximal Cohen-Macaulay if \(\mathrm{depth}_A(M)=\dim(A)\)).
When the ring \(A\) is particularly well-behaved, MCM-modules correspond to familiar objects. For instance, when \(A\) is reduced of Krull dimension one, MCM A-modules are exactly the torsion free modules. Analogously, in the case of \(A\) being a normal surface singularity, MCM-modules correspond to the reflexive \(A\)-modules.
The study of MCM-modules rises from the theory of integral representations of finite groups and has revealed to own many connections with other fields, as it can be the theory of vector bundles on projective curves, McKay correspondence for the finite subgroups of \(\mathrm{SL}_2(\mathbb{C})\), etc.
A breakthrough in this area was the classification of hypersurface singularities \((R/f,m)\) which support only a finite number of non-isomorphic indecomposable MCM-modules: they correspond exactly to simple hypersurface singularities \(A_n, D_n,E_6, E_7\) and \(E_8\) [\textit{R. O. Buchweitz} et al., Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)]. Moreover, in the aforementioned paper two limiting non-isolated hypersurface singularities with a countable number of non-isomorphic MCM-modules were identified: \(A_{\infty}\) and \(D_{\infty}\).
\textit{F.-O. Schreyer} [Lect. Notes Math. 1273, 9--34 (1987; Zbl 0719.14024)] whether a non-isolated Cohen-Macaulay surface singularity \((A,m)\) of countable CM-type should be isomorphic to \(A\cong B^G\) which \(B\) either \(A_{\infty}\) or \(D_{\infty}\) and \(G\) a finite group of automorphisms of \(B\).
The monography under review deals with this set of problems. In particular, among the central results, it is exhibited in Theorem \(10.6\) a counterexample to Schreyer's conjecture. The authors also deal with rings of greater CM-complexity. For instance, it is shown that degenerate cusps, like for instance the ordinary triple point \(\mathbb{K}[[x,y,z]]/xyz\), as they were introduced by \textit{N. I. Shepherd-Barron} [Prog. Math. 29, 33--84 (1983; Zbl 0506.14028)], are of tame CM-type (see Definition \(8.13\) and Theorem \(8.15\)).
The technical core of this monography is the construction, associated to a non normal, reduced CM surface singularity \(A\), of the categories \(\mathrm{Tri}(A)\) (category of triples) and \(\mathrm{Rep}(\chi_A)\) (category of elements of a certain bimodule \(\chi_A\) and of two functors
\[
CM(A)\stackrel{F}{\rightarrow} \mathrm{Tri}(A)\stackrel{H}{\rightarrow} \mathrm{Rep}(\chi_A)
\]
such that F is an equivalence and H preserves indecomposibility and isomorphims (see Theorem 3.5 and Proposition 8.9). Therefore, the description of indecomposable MCM-modules is reduced to a problem of linear algebra (which the authors called ``a matrix problem''). Maximal Cohen-Macaulay modules; matrix factorizations; non-isolated surface singularities; degenerate cusps; tame matrix problems Cohen-Macaulay modules in associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Cohen-Macaulay modules, Singularities of surfaces or higher-dimensional varieties Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems | 0 |
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 paper under review, the authors study the two-dimensional log-canonical pairs \((X,\Delta)\) such that \(K_X +\Delta\) is an ample Cartier divisor with \((K_X+\Delta)^2 =1\), giving some applications to stable surfaces. In the first part, they prove the following classification result. Let \((X,\Delta)\) be a log-canonical pair of dimension two with \(\Delta>0\), \(K_X+\Delta\) Cartier and ample, and \((K_X+\Delta)^2=1\).
Then \((X, \Delta)\) belongs to one of the following four types:
(\(P\)) \(X= \mathbb{P}^2\) and \(\Delta\) is a nodal quartic.
(\(dP\)) \(X\) is a (possibly singular) del Pezzo surface of degree 1 and the curve \(\Delta\) belongs to \(|-2K_X|\).
(\(E_-\)) Let \(E\) be an elliptic curve, let \(a: \widetilde X=:\mathbb{P}(\mathcal O_E\oplus \mathcal O_E(-x)) \to E\) where \(x\in E\) is a point and let \(C_0\) be the only curve in \(|\mathcal O_{\widetilde X} (1)|\), \(C_0^2=-1\). Set \(F=a^{-1}(x)\); then the normal surface \(X\) is obtained from \(\widetilde X\) contracting \(C_0\) to an elliptic Gorenstein singulaity of degree 1. Here \(\Delta \) is the image of a curve \(\Delta_0 \in |2(C_0+F)| \) disjoint from \(C_0\).
(\(E_+\)) \(X=S^2E,\) where \(E\) is an elliptic curve. Let \(a:X\to E\) be the Albanese map, \(F\) the class of a fiber of \(a\) and \(C_0\) the image in \(X\) of the curve \(\{0\}\times E + E\times \{0\}\), so that \(C_0F=C_0^2=1\). Then \(\Delta\) is a divisor numerically equivalent to \(3C_0-F\).
In the second part of the paper, the authors give some applications to stable surfaces. Using a result of [\textit{J. Kollár}, Singularities of the minimal model program. With the collaboration of Sándor Kovács. Cambridge: Cambridge University Press (2013; Zbl 1282.14028)], they prove that if \(X\) is a Gorenstein stable surface with \(K_X^2=1\), then its normalization \((\overline{X}, \overline{D})\) is one of the four types above. They also give the possible values of \(\chi(X)\) for each type, showing in particular that there are no Gorenstein stable surfaces with \(K_X^2=1\) and \(\chi(X) < 0\).
In the last section the case \(\Delta=0\) is considered. The authors give a rough classification of Gorenstein log-canonical surfaces with \(K_X\) ample and \(K_X^2=1\), according to the Kodaira dimension of the minimal desingularization \(\widetilde X \) of \(X\). log-canonical pair; stable surface; geography of surfaces Franciosi, M; Pardini, R; Rollenske, S, Log-canonical pairs and Gorenstein stable surfaces with \(K_X^2=1\), Compos. Math., 151, 1529-1542, (2015) Families, moduli, classification: algebraic theory, Surfaces of general type Log-canonical pairs and Gorenstein stable surfaces with \(K_X^2=1\) | 0 |
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 Hilbert scheme \(\text{Hilb}^{p(m)}(\mathbb P ^n)\) parametrizes closed subschemes of \(\mathbb P^n\) with given Hilbert polynomial \(p(m)\). By a result of Groethendieck, it is known that \(\text{Hilb}^{p(m)}(\mathbb P ^n)\) is a projective scheme and by a result of Hartshorne, it is connected. In the paper under review, the authors study the component \(H_n\) of the Hilbert scheme whose general point parametrizes a pair of codimension two linear subspaces of \(\mathbb P ^n\) for \(n\geq 3\). In particular they show that \(H_n\) is smooth and isomorphic to the blow up of the symmetric square of \(\mathbb G (n-2,n)\) along the diagonal, and that \(H_n\) intersects only one other component of \(\text{Hilb}^{p(m)} (\mathbb P ^n)\). Moreover, they show that \(H_n\) is a Mori dream space. Hilbert scheme; Mori dream space; Stable base locus decomposition D. Chen, I. Coskun & S. Nollet, Hilbert scheme of a pair of codimension two linear subspaces. Comm. Algebra 39, no. 8, 3021--3043, 2011.arXiv:0909.5170 Rational and birational maps, Minimal model program (Mori theory, extremal rays), Grassmannians, Schubert varieties, flag manifolds, Fine and coarse moduli spaces, Parametrization (Chow and Hilbert schemes) Hilbert scheme of a pair of codimension two linear subspaces | 0 |
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 describe how certain standard opens \(\mathrm{Hilb}^\beta\) of the Hilbert scheme \(\mathrm{Hilb}^N_{\mathbf A^n_A}\) parameterizing \(N\) points of \(\mathbf A^n_A\) are embedded into Grassmannians. These open subsets of the Hilbert scheme are the intersection of a corresponding open affine of the Grassmannian and a closed stratum determined by a Fitting ideal. As an application, the authors give a cover for the scheme parameterizing \(0\)-dimensional closed subschemes in \(\mathbb P^n_k\), \(k\) an algebraically closed field, that are flat, finite of relative rank \(n + 1\), and non-degenerate. The authors also show that the ideal generated by the commutator relations, often used in literature in the constructions of the local open schemes \(\mathrm{Hilb}^\beta\), equals the Fitting ideal arising from the graded, global situation. As an application, the authors give a new proof of the following result, already proved by \textit{K. Ranestad} and \textit{F.-O. Schreyer} [J. Reine Angew. Math. 525, 147--181 (2000; Zbl 1078.14506)]: the scheme \(\mathrm{VPS}^{n+1}_Z\) of length \(n+1\) subschemes in \(\mathbb P^n\) apolar to the annihilator scheme \(Z\) of a smooth quadratic surface is closed in the Grassmannian of rank \(n+1\) quotients of the vector space of two-forms on projective \(n\)-space. In the Appendix, the authors restate some of their results on ideals and Hilbert schemes for modules and Quot schemes. fitting ideals; Hilbert schemes; quot schemes; Grassmannian; strongly generated; apolarity Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Varieties of low degree, Projective and free modules and ideals in commutative rings, Grassmannians, Schubert varieties, flag manifolds, Modules, bimodules and ideals in associative algebras Explicit projective embeddings of standard opens of the Hilbert scheme of points | 0 |
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 studies the analytic structure on the moduli space \(M_{g/{\mathbb R}}\) of real algebraic curves of genus \(g\). A connected component of \(M_{g/{\mathbb R}}\) can be viewed as a semianalytic variety being a quotient of a real analytic Teichmüller manifold \(T(X)\) of marked real algebraic curves modeled on a nonsingular real algebraic curve \(X\) by the properly discontinuous action of the real modular group \(\text{Mod}(X)\) on \(T(X)\). The main result of the paper is that any component of \(M_{g/{\mathbb R}}\) has a nonempty boundary, hence is not an analytic variety. moduli spaces of real algebraic curves; real analytic Teichmüller manifold; real quotient singularities; real analytic manifolds; proper discontinuous group action; semianalytic varieties Huisman, Real quotient singularities and nonsingular real algebraic curves in the boundary of the moduli space, Compos. Math. 118 pp 42-- (1999) Families, moduli of curves (analytic), Singularities of curves, local rings, Local complex singularities, Real-analytic manifolds, real-analytic spaces, Real-analytic and semi-analytic sets Real quotient singularities and nonsingular real algebraic curves in the boundary of the moduli space | 0 |
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 del Pezzo surface is a surface whose anticanonical divisor is ample. It is known that the only del Pezzo surfaces are \(\mathbb{P}^1\times \mathbb{P}^1\) and the blow up of \(\mathbb{P}^2\) in \(r\leq 8\) points in general position. The degree of a del Pezzo surface is \(9-r\).
The main result in the paper under review says that if \(V\) is a variety of dimension 3 with mild singularities, with Picard number 2, fibred into del Pezzo surfaces of degree 2 and satisfying the \(K^2\)-condition and \(V\) is birationally equivalent to a Mori fibre space \(Y\to B\), then the Mori fibre space structure on \(Y\) is induced by the fibration on \(V\) into del Pezzo surfaces.
This extends a result in [\textit{A. V. Pukhlikov}, Dokl. Math. 58, No. 1, 8--10 (1998; Zbl 0985.14006)] where the fibration is smooth.
Here by ``mild singularities'' we mean that the singularities are isolated and of type \(\mathbb{C}^3/\langle-Id\rangle\) and that the fibres containing singularities are quartic cones in \(\mathbb{P}(1,1,1,2)\).
A variety \(V\) of dimension 3 fibred into surfaces satisfies the \(K^2\)-condition if for all \(m\in\mathbb{Z}_{>0}\) and for every curve \(f\) contained in a fibre, the cycle \(mK_V^2-f\) is not numerically equivalent to an effective cycle.
The result says, roughly speaking, that there is only one Mori fibre space structure on \(V\). A variety with this property cannot be rational. As a corollary of his main result the author proves the non-rationality of some specific varieties fibred into del Pezzo surfaces, which carry a regular action of the group \(\mathrm{PSL}_2(7)\). del Pezzo; rationality; Mori fibre space; maximal singularities; subgroups of Cremona group Krylov, I., Birational geometry of del Pezzo fibrations with terminal quotient singularities, (2016) Rationality questions in algebraic geometry, \(3\)-folds, Birational automorphisms, Cremona group and generalizations Birational geometry of del Pezzo fibrations with terminal quotient singularities | 0 |
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 give an explicit geometric description to some of H. Nakajima's quiver varieties. More precisely, let \(X = {\mathbb{C}}^2\), \(\Gamma \subset \text{SL}({\mathbb{C}}^2)\) be a finite subgroup, and \(X_\Gamma\) be a minimal resolution of \(X/\Gamma\). The main result states that \(X^{\Gamma [n]}\) (the \(\Gamma\)-equivariant Hilbert scheme of \(X\)) and \(X_\Gamma^{[n]}\) (the Hilbert scheme of \(X_\Gamma\)) are quiver varieties for the affine Dynkin graph corresponding to \(\Gamma\) via the McKay correspondence with the same dimension vectors but different parameters.
In section two, basic concepts such as the definition of quivers, quiver varieties, representation of quivers and the construction of Crawley-Boevey were reviewed. In section three, the author reproduced in a short form a geometric version of the McKay correspondence based on investigation of \(X_\Gamma\), and proved a generalization of certain result of \textit{M. Kapranov} and \textit{E. Vasserot} [Math. Ann. 316, No. 3, 565--576 (2000; Zbl 0997.14001)]. The main result mentioned above was verified in section four. In particular, it follows that the varieties \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) are diffeomorphic. In section five, \(({\mathbb{C}}^* \times {\mathbb{C}}^*)\)-actions on \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) for cyclic \(\Gamma \cong {\mathbb{Z}}/d {\mathbb{Z}}\) were considered. The author proved the combinatorial identity \(UCY(n, d) = CY(n, d)\) where \(UCY\) and \(CY\) denote the number of uniformly colored diagrams and the number of collections of diagrams respectively. quiver varieties; Hilbert schemes; McKay correspondence; moduli space Kuznetsov, A.: Quiver varieties and Hilbert schemes. Moscow Math. J. \textbf{7}, 673-697 (2007). arXiv:math.AG/0111092 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Quiver varieties and Hilbert schemes. | 0 |
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 \(F\) denote a binary form of order \(d\) over the complex numbers. If \(r\) is a divisor of \(d\) , then the Hilbert covariant \(H_{r,d} (F) \) vanishes exactly when \(F\) is the perfect power of an order \(r\) form. In geometric terms, the coefficients of \(H\) give defining equations for the image variety \(X\) of an embedding \(P^r \hookrightarrow P^d\). In this paper the authors describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on \(X\). They prove that the ideal generated by the coefficients of \(H\) defines \(X\) as a scheme. Finally, they exhibit a generalisation of the Gottingen covariants to \(n\)-ary forms using the classical Clebsch transfer principle. binary forms; covariants; \(SL_2\)-representations A. Abdesselam and J. Chipalkatti. On Hilbert covariants. Canad. J. Math., 66(1):3-- 30, 2014. Actions of groups on commutative rings; invariant theory, Geometric invariant theory On Hilbert covariants | 0 |
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 aim of this article is to study the structure of \(H^2 (X,K_2)\) of a surface \(X\) with prescribed singularities.
Let \(X'\) be a smooth projective surface over \(\mathbb{C}\), \(m\) an effective divisor on \(X'\) and \(\pi : X' \to X\) such that \(\pi : m \to S\) is a finite surjective map between reduced curves and \(U = X' \backslash m \simeq X \backslash S\). The author studies the relations between the group \(H^2 (X,K_2)\), the 1-motive \(J^2 (X)\) defined by \(J^2 (X) = H^3 (X,\mathbb{C})/F^2 H^3 (X, \mathbb{C}) + H^3 (X, \mathbb{Z})\) and \(G_{um}\) the generalized albanese variety of \(X'\) with modulus \(m : G_{um} (\mathbb{C}) \simeq H^0 (X', \Omega_{X'} (m))^*_{d = 0}/H_2 (U, \mathbb{Z})\); the generalized albanese map \(\alpha_{um} : U \to G_{um} (\mathbb{C})\) is given by \(\alpha_{um} (x)=(\int_\gamma \omega_1, \ldots, \int_\gamma \omega_n)\) modulo periods, for any path \(\gamma\) joining a fixed point \(x_0\) to \(x\) and where \((\omega_i)\) is a basis of \(H^0 (X', \Omega_{X'} (m))_{d = 0}\). The author gets the following results:
(i) There is a surjective homomorphism \(G_{um} (\mathbb{C}) \to J^2 (X)\) which is an isomorphism if \(S\) is integral.
(ii) If \(H^2 (X', {\mathcal O}_{X'}) = 0\) then \(J^2 (X)\) is an extension of \(\text{Alb}_{X'}\) by a torus of dimension \(d = \text{rank(NS}(m)/(\text{NS}(S) +\text{NS}(X')))\) where NS\(( )\) is the Néron-Severi group.
(iii) \(\alpha_{um}\) induces a surjective homomorphism \(H^2 (X, K_2)_0 \to J^2 (X)\) where \(H^2 (X,K_2)_0\) is the group of zero cycles of degree zero.
To prove the first result, the author considers the following diagram:
\[
\begin{matrix} H^2 (X, \mathbb{C}) & \longrightarrow & H^2 (S, \mathbb{C}) & \longrightarrow & H^3_c (U, \mathbb{C}) & \longrightarrow & H^3 (X, \mathbb{C}) & \longrightarrow & 0 \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \pi^* \\ H^2 (X', \mathbb{C}) & \longrightarrow & H^2 (m, \mathbb{C}) & \longrightarrow & H^3_c (U, \mathbb{C}) & \longrightarrow & H^3 (X', \mathbb{C}) & \longrightarrow & 0 \end{matrix}
\]
By Hodge theory, we have \(F^2 H^3 (X, \mathbb{C}) \cap W^2 H^3 (X, \mathbb{C}) = F^2 H^3 (X, \mathbb{C}) \cap \text{Ker} (\pi^*) = 0\), and we get \(F^2 H^3 (X, \mathbb{C}) = F^2 H^3 (X', \mathbb{C}) = H^1 (X', \Omega^2)\). Then we deduce from the isomorphism \(H^1 (U, \mathbb{C}) \simeq \Omega^{\text{inv}}_{G_{um}} \oplus H^1 (X', \Omega^2)\) a surjective map \(\Omega^{\text{inv} *}_{G_{um}} \to H^3 (X, \mathbb{C})/F^2 H^3 (X, \mathbb{C})\), and taking quotients respectively by \(H^3_c (U, \mathbb{Z}) \simeq H_1 (U, \mathbb{Z})\) and \(H^3 (X, \mathbb{Z})\) we obtain the surjective map \(G_{um} (\mathbb{C}) \to J^2 (X)\). If \(S\) is integral \(H^2 (S, \mathbb{C}) \simeq \mathbb{C}\), then \(H^3_c (U, \mathbb{C}) \simeq H^3 (X, \mathbb{C})\), and the second part of the first statement follows. We obtain the second statement from the exact sequence:
\[
0 \to {H^2 (m, \mathbb{C}) \over H^2 (S, \mathbb{C}) + H^2 (X', \mathbb{C})} \to {H^3_c (U, \mathbb{C}) \over H^2 (S, \mathbb{C}) + H^1 (X', {\mathcal O}_{X'})} \to {H^3 (X', \mathbb{C}) \over F^2 H^3 (X', \mathbb{C})} \to 0
\]
and from the isomorphism \(H^3_c (U, \mathbb{C})/(H^2 (S, \mathbb{C}) + H^1 (X', {\mathcal O}_{X'})) \simeq H^3 (X', \mathbb{C})/F^2 H^3 (X', \mathbb{C})\). To obtain the last statement, we have to look at the decomposition in irreducible components of the divisors \(S\) and \(m\) and to study the map \(H^2 (S, \mathbb{C}) \to H^3_c (U, \mathbb{C})\). Albanese variety; algebraic cycles; second cohomology group of a surface with prescribed singularities; curves on surfaces; effective divisor; 1- motive; periods (Co)homology theory in algebraic geometry, Generalizations (algebraic spaces, stacks), Algebraic cycles, Singularities of surfaces or higher-dimensional varieties, Curves in algebraic geometry On glueing curves on surfaces and zero cycles | 0 |
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 article consists of two parts. The first part is a survey on the normal reduction numbers of normal surface singularities. It includes results on elliptic singularities, cone-like singularities and homogeneous hypersurface singularities. In the second part, we prove a new results on the normal reduction numbers and related invariants of Brieskorn complete intersections. normal reduction number; normal surface singularity; geometric genus; elliptic singularity; Brieskorn complete intersection; homogeneous hypersurface singularity Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Complex surface and hypersurface singularities, Integral closure of commutative rings and ideals Normal reduction numbers of normal surface singularities | 0 |
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 \(G_0\) be a split real adjoint simple Lie group and \(S(G_0)\) the associated symmetric space (i.e., the space of Cartan involutions of \(G_0\)). We recall that, given a connected oriented closed surface \(\Sigma\) of genus \(g\geq 2\) and the corresponding character variety \(\text{Hom}(\pi_1(\Sigma), G_0)/G_0\) of representations from \(\pi_1(\Sigma)\) to \(G_0\), in [Topology 31, No. 3, 449--473 (1992; Zbl 0769.32008)] \textit{N. J. Hitchin} introduced the fundamental notion of Teichmüller component, now called \textit{Hitchin component}, which is a special connected component \(\mathcal H(\Sigma, G_0)\) of \(\text{Hom}(\pi_1(\Sigma), G_0)/G_0\) that contains the connected component of \(\text{Hom}(\pi_1(\Sigma), \text{PSL}(2, \mathbb R))/\text{PSL}(2,\mathbb R)\) corresponding to the Teichmüller space of \(\Sigma\). We also recall that the elements of the Hitchin component \(\mathcal H(\Sigma, G_0)\), the so-called \textit{Hitchin representations}, are the monodromies of the flat connections \(\nabla + \Phi + \Phi^*\) of some special Higgs bundles \((\mathcal G, \Phi)\) over \(\Sigma\), in which \(\nabla\) admits a real structure \(\rho\) so that the pair \((\nabla, \rho)\) is solution to the self-duality equations of \((\mathcal G, \Phi)\). The special Higgs bundles \((\mathcal G, \Phi)\) to be considered are parametrised by families of holomorphic differentials \(q = (q_1, \ldots, q_\ell)\), \(\ell = \text{rank}(G_0)\), of degrees \((m_1+1, \dots, m_\ell +1)\), respectively, for some integers \(m_i\) uniquely determined by the Lie group structure of \(G_0\).
The main result of this paper is the following: When \(\text{rank}(G_0) = 2\) (i.e., \(G_0 = \text{SL}(3, \mathbb R)\), \(\text{PSp}(4, \mathbb R)\) or \(\text{G}_{2,0}\)) for each Hitchin representation \(\delta \in \mathcal H(\Sigma, G_0)\), there exists a uniquely associated \(\delta\)-equivariant minimal surface \(\phi^\delta: \Sigma \to S(G_0)\) in the symmetric space \(S(G_0)\) of \(G_0\). Such map \(\phi^\delta\) has the form \(\phi^\delta = p\circ f^\delta\), where \(f^\delta: \Sigma \to \mathsf X\) is an appropriate holomorphic map, called \textit{cyclic surface}, into the space \(\mathsf X\) of Hitchin-Kostant quadruples, and \(p: \mathsf X \to S(G_0)\) is the natural projection of \(\mathsf X\) onto \(S(G_0)\).
Note that the existence part of this theorem was proved by the author in [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 3, 439--471 (2008; Zbl 1160.37021)] without any assumption on the rank. The uniqueness part is now reached by first proving a general criterion for infinitesimal rigidity of cyclic surfaces and then exploiting previous results of the author on the properness of the energy functionals on Teichmüller spaces determined by Hitchin representations.
Using the Hitchin parametrisation of the space of minimal surfaces, the above result also yields that for each \(G_0\) as above, there exists an equivariant diffeomorphism between the Hitchin component \(\mathcal H(\Sigma, G_0)\) and the space of pairs \((J, Q)\), given by a complex structure \(J\) on \(\Sigma\) and a \(J\)-holomorphic differential \(Q\) of degree \(\frac{\dim(G_0) - 2}{2}\).
Combining all this with the theory of positive bundles and the results of \textit{B. Berndtsson} [Ann. Math. (2) 169, No. 2, 531--560 (2009; Zbl 1195.32012)], the author also obtains that for \(G_0\) as above, the Hitchin component \(\mathcal H(\Sigma, G_0)\) carries a complex structure and a \(1\)-dimensional family of compatible mapping class group invariant Kähler metrics, for which the Fuchsian locus is totally geodesic and whose restriction to the Fuchsian locus is the Weil-Petersson metric. The paper is then concluded with an Area Rigidity Theorem for Hitchin components. Teichmüller spaces; Hitchin components; Hitchin representations; minimal surfaces; Higgs bundles Labourie, F.: Cyclic surfaces and Hitchin components in rank 2. arXiv:1406.4637 (2014) Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Vector bundles on curves and their moduli, Complex-analytic moduli problems, Differential geometry of symmetric spaces, Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Cyclic surfaces and Hitchin components in rank 2 | 0 |
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 celebrated theorem due to \textit{O. Zariski} [Ann. Math. 40, 639--689 (1939; Zbl 0021.25303)] asserts that if \(S\) is an algebraic surface over a field (algebraically closed and of characteristic zero), one can resolve the singularities of \(S\) by a finite sequence of point blow-ups followed by normalizations (called normalized blow-ups). We give a new approach to prove this theorem by attaching to a germ \((S,0)\) of a complex normal surface a pair \((\nu,\gamma)\) of natural numbers, where \(\nu=e(S,0)\) is the multiplicity of the germ, and \(\gamma=e (\Delta_p,0)\) is the multiplicity of the discriminant \(\Delta_p\) of a generic projection from a small representative of \((S,0)\) onto an open subset \(U\) of \(\mathbb{C}^2\). We prove that after a finite number \(N_S\) of normalized blow-ups, all the singular points \(0_i\) on the surface \(S_{N_S}\) obtained, have pairs \((\nu_i,\gamma_i)\) strictly smaller (for the lexicographic order) than the original pair \((\nu, \gamma)\) for \((S,0)\). This implies the theorem by Zariski, since \(\nu =1\) or \(\gamma=1\) for a germ both yield that the corresponding germ is smooth. Moreover, we have the following bound for the number \(N_S\): \(N_S\) is not greater than the number of point blow-ups necessary to get an embedded resolution of the discriminant curve \(\Delta_p\subset U\) considered above. blow-ups; normalizations; embedded resolution Bondil, R.; Lê, D. T.: Résolution des singularités de surfaces par éclatements normalisés. Trends in singularities, 31-81 (2002) Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Resolution of surface singularities by normalized blow-ups. (Multiplicity, polar multiplicity, and minimal singularities) | 0 |
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 non-hyperelliptic smooth projective curve of genus \(g\geq 3\) over the complex field. Let \(\Theta\) denote the Riemann theta divisor on \(\text{Pic}^{g-1}(C)\) and \(\Theta_0\) the symmetric theta divisor on the Jacobian variety \(JC\) , obtained translating \(\Theta\) by a theta characteristic. \textit{J. Fay} [``Theta functions on Riemann surfaces'', Lect. Notes Math. 352 (1973; Zbl 0281.30013)] observed that, if \(D\) is a divisor in the linear system \(|2\Theta_0|\), then mult\(_0D\geq 4\) if and only if \(D\) contains the surface (difference) \(C-C\). Moreover, if we denote by \(\Gamma_0\) the hyperplane in \(H^0(JC, {\mathcal O}(2\Theta_0))\) of 2\(\theta\)-divisors containing \({\mathcal O}\), and \(\Gamma_{00}\) its subspace of those having multiplicity at least 4 at \({\mathcal O}\), then \(\Gamma_0/\Gamma_{00}\simeq \text{Sym}^2H^0(K)\).
In the paper under review, the authors study the following subseries of \({\mathbb P}\Gamma_{00}\): \({\mathbb P}\Gamma_{11} =\{D\in{\mathbb P}\Gamma_{00} \mid C_2-C_2\subset D\}\), where \(C_2\) is the second symmetric power of \(C\); \({\mathbb P}\Gamma_{000}\) formed by divisors having multiplicity \(\geq 6\) at \({\mathcal O}\); \({\mathbb P}\Gamma_{00}^{(2)}\) formed by divisors \(D\) that are singular along \(C-C\). In particular they prove the existence of a filtration
\[
\Gamma_{11} \subset \Gamma_{000} \subset \Gamma_{00}^{(2)} \subset \Gamma_{00} \subset \Gamma_0
\]
whose quotients are as follows: \(\Gamma_{00}/\Gamma_{00}^{(2)}\simeq \wedge^3H^0(K)\); if \(C\) is non-trigonal, then \(\Gamma_{00}^{(2)}/\Gamma_{11} \simeq \text{Sym}^2I(2)\), where \(I(2)\) is the space of the quadrics in the ideal of the canonical image of \(C\); \(\Gamma_{000}/\Gamma_{11} \simeq \operatorname {Ker}m\), where \(m\) is the multiplication map \(m: \text{Sym}^2I(2)\to I(4)\). theta functions; jacobian; canonical curve; vector bundle C. Pauly - E. Previato, Singularities of 2\(\theta \)-divisors in the Jacobian, Bull. Soc. Math. France 129 (2001), 449-485. Zbl1016.14013 MR1881203 Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli, Special divisors on curves (gonality, Brill-Noether theory), Theta functions and abelian varieties Singularities of \(2\Theta\)-divisors in the jacobian | 0 |
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 Gorenstein smoothable singularity with \(p_g(X,x)\geq 1\). It is proved that
\[
\delta_2(X,x)=h^1(S)+\mu(X,x)-\tau(X,x)-p_g(X,x)+1\;.
\]
Here \(\mu(X,x)\) and \(\tau(X,x)\) are the generalized Milnor number and Tjurina number of \((X,x), \delta_2(X,x)\) is the second pluri-genus and \(p_g(X,x)\) the geometric genus. \(f:(M, E)\to (X,x)\) is the minimal good resolution, \(E\) the exceptional divisor, and \(S=(\Omega_M^1(\log E))^\ast\). This is a generalization of the corresponding result of Okuma, for complete intersection isolated surface singularities. second pluri-genus; smoothable; Gorenstein; Milnor number; Tjurina number Du R, Yau S T. The second pluri-genus of smoothable Gorenstein surface singularities. Sci China Math, 2010, 53: 635--639 Invariants of analytic local rings, Local complex singularities, Deformations of complex singularities; vanishing cycles, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Deformations of singularities The second pluri-genus of smoothable Gorenstein surface singularities | 0 |
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 stratification associated with the number of generators of the ideals of the punctual Hilbert scheme of points on the affine plane has been studied since the 1970s. In this paper, we present an elegant formula for the E-polynomials of these strata. Symmetric functions and generalizations, Polynomial rings and ideals; rings of integer-valued polynomials, Parametrization (Chow and Hilbert schemes) A note on the E-polynomials of a stratification of the Hilbert scheme of points | 0 |