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100 | Higgs line bundles, Green-Lazarsfeld sets,and maps of Kähler manifolds to curves | math.AG | Let $X$ be a compact K\"ahler manifold. The set $\cha(X)$ of one-dimensional
complex valued characters of the fundamental group of $X$ forms an algebraic
group. Consider the subset of $\cha(X)$ consisting of those characters for
which the corresponding local system has nontrivial cohomology in a given
degree $d$. This set is shown to be a union of finitely many components that
are translates of algebraic subgroups of $\cha(X)$. When the degree $d$ equals
1, it is shown that some of these components are pullbacks of the character
varieties of curves under holomorphic maps. As a corollary, it is shown that
the number of equivalence classes (under a natural equivalence relation) of
holomorphic maps, with connected fibers, of $X$ onto smooth curves of a fixed
genus $>1$ is a topological invariant of $X$. In fact it depends only on the
fundamental group of $X$. | math |
101 | A theory of algebraic cocycles | math.AG | We introduce the notion of an algebraic cocycle as the algebraic analogue of
a map to an Eilenberg-MacLane space. Using these cocycles we develop a
``cohomology theory" for complex algebraic varieties. The theory is bigraded,
functorial, and admits Gysin maps. It carries a natural cup product and a
pairing to $L$-homology. Chern classes of algebraic bundles are defined in the
theory. There is a natural transformation to (singular) integral cohomology
theory that preserves cup products. Computations in special cases are carried
out. On a smooth variety it is proved that there are algebraic cocycles in each
algebraic rational $(p,p)$-cohomology class. | math |
102 | Zariski Geometries | math.AG | We characterize the Zariski topologies over an algebraically closed field in
terms of general dimension-theoretic properties. Some applications are given to
complex manifold and to strongly minimal sets. | math |
103 | Configuration spaces and the space of rational curves on a toric variety | math.AG | The space of holomorphic maps from $S^2$ to a complex algebraic variety $X$,
i.e. the space of parametrized rational curves on $X$, arises in several areas
of geometry. It is a well known problem to determine an integer $n(D)$ such
that the inclusion of this space in the corresponding space of continuous maps
induces isomorphisms of homotopy groups up to dimension $n(D)$, where $D$
denotes the homotopy class of the maps. The solution to this problem is known
for an important but special class of varieties, the generalized flag
manifolds: such an integer may be computed, and $n(D)\to\infty$ as
$D\to\infty$. We consider the problem for another class of varieties, namely,
toric varieties. For smooth toric varieties and certain singular ones, $n(D)$
may be computed, and $n(D)\to\infty$ as $D\to\infty$. For other singular toric
varieties, however, it turns out that $n(D)$ cannot always be made arbitrarily
large by a suitable choice of $D$. | math |
104 | Stable vector bundles on algebraic surfaces | math.AG | We prove an existence result for stable vector bundles with arbitrary rank on
an algebraic surface, and determine the birational structure of certain moduli
space of stable bundles on a rational ruled surface. | math |
105 | Toric Intersection Theory for Affine Root Counting | math.AG | Given any polynomial system with fixed monomial term structure, we give
explicit formulae for the generic number of roots with specified coordinate
vanishing restrictions. For the case of affine space minus an arbitrary union
of coordinate hyperplanes, these formulae are also the tightest possible upper
bounds on the number of isolated roots. We also characterize, in terms of
sparse resultants, precisely when these upper bounds are attained. Finally, we
reformulate and extend some of the prior combinatorial results of the author on
which subsets of coefficients must be chosen generically for our formulae to be
exact.
Our underlying framework provides a new toric variety setting for
computational intersection theory in affine space minus an arbitrary union of
coordinate hyperplanes. We thus show that, at least for root counting, it is
better to work in a naturally associated toric compactification instead of
always resorting to products of projective spaces. | math |
106 | On Hyper Kähler manifolds associated to Lagrangean Kähler submanifolds of $T^*{\Bbb C}^n$ | math.AG | For any Lagrangean K\"ahler submanifold $M \subset T^*{\Bbb C}^n$, there
exists a canonical hyper K\"ahler metric on $T^*M$. A K\"ahler potential for
this metric is given by the generalized Calabi Ansatz of the theoretical
physicists Cecotti, Ferrara and Girardello. This correspondence provides a
method for the construction of (pseudo) hyper K\"ahler manifolds with large
automorphism group. Using it, a class of pseudo hyper K\"ahler manifolds of
complex signature $(2,2n)$ is constructed. For any hyper K\"ahler manifold $N$
in this class a group of automorphisms with a codimension one orbit on $N$ is
specified. Finally, it is shown that the bundle of intermediate Jacobians over
the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper
K\"ahler metric of complex signature $(2,2n)$. | math |
107 | Rational curves and ampleness properties of the tangent bundle of algebraic varieties | math.AG | The purpose of this paper is to translate positivity properties of the
tangent bundle (and the anti-canonical bundle) of an algebraic manifold into
existence and movability properties of rational curves and to investigate the
impact on the global geometry of the manifold $X$. Among the results we prove
are these:
\quad If $X$ is a projective manifold, and ${\cal E} \subset T_X$ is an ample
locally free sheaf with $n-2\ge rk {\cal E}\ge n$, then $X \simeq \EP_n$.
\quad Let $X$ be a projective manifold. If $X$ is rationally connected, then
there exists a free $T_X$-ample family of (rational) curves. If $X$ admits a
free $T_X$-ample family of curves, then $X$ is rationally generated. | math |
108 | Quantum cohomology of projective bundles over P^n | math.AG | In this paper, we attempt to determine the quantum cohomology of projective
bundles over the projective space P^n. In contrast to the previous examples,
the relevant moduli spaces in our case frequently do not have expected
dimensions. It makes the calculation more difficult. We overcome this
difficulty by using excessive intersection theory. | math |
109 | Extensions of vector bundles and rationality of certain moduli spaces of stable bundles | math.AG | In this paper, it is proved that certain stable rank-3 vector bundles can be
written as extensions of line bundles and stable rank-2 bundles. As an
application, we show the rationality of certain moduli spaces of stable rank-3
bundles over the projective plane P^2. | math |
110 | Rank-3 stable bundles on rational ruled surfaces | math.AG | In this paper, we compare the moduli spaces of rank-3 vector bundles stable
with respect to different ample divisors over rational ruled surfaces. We also
discuss the irreducibility, unirationality, and rationality of these moduli
spaces. | math |
111 | Topological arrangement of curves of degree 6 on cubic surfaces in $\Bbb R P^3$ | math.AG | A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R
P^3$. The papers classifies the nonsingular curves cut in this way on
non-singular cubic surfaces up to homeomorphism.
Two issues new in the study related to the first part of the 16th Hilbert
problem appear in this classification. One is the distribution of the
components of the curve between the components of the non-connected cubic
surface which turns out to depend on the patterns of arrangements (see Theorem
1). The other is presence of positive genus among the components of the
complement and genus-related restrictions (see Theorems 3 and 4). | math |
112 | Adjunction inequality for real algebraic curves | math.AG | The zero set of a real polynomial in two variable is a curve in $\mathbb
R^2$. For a generic choice of its coefficients this is a non-singular curve, a
collection of circles and lines properly embedded in $\mathbb R^2$. What
topological arrangements of these circles and lines appear for the polynomials
of a given degree? This question arised in the 19th century in the works of
Harnack and Hilbert and was included by Hilbert into his 16th problem. Several
partial results were obtained since then. However the complete answer is known
only for polynomials of degree 5 or less. The paper presents a new partial
result toward the solution of the 16th Hilbert problem.
The proof makes use of the proof by Kronheimer and Mrowka of the Thom
conjecture in $\mathbb C P^2$. | math |
113 | Subspace Arrangements over Finite Fields: Cohomological and Enumerative Properties | math.AG | The enumeration of points on (or off) the union of some linear or affine
subspaces over a finite field is dealt with in combinatorics via the
characteristic polynomial and in algebraic geometry via the zeta function. We
discuss the basic relations between these two points of view. Counting points
is also related to the $\ell$-adic cohomology of the arrangement (as a
variety). We describe the eigenvalues of the Frobenius map acting on this
cohomology, which corresponds to a finer decomposition of the zeta function.
The $\ell$-adic cohomology groups and their decomposition into eigenspaces are
shown to be fully determined by combinatorial data. Finally, it is shown that
the zeta function is determined by the topology of the corresponding complex
variety in some important cases. | math |
114 | On Rigidity and the Albanese Variety for Parallelizable Manifolds | math.AG | We study the rigidity questions and the Albanese Variety for Complex
Parallelizable Manifolds. Both are related to the study of the cohomology group
$H^1(X,\mathcal O)$. In particular we show that a compact complex
parallelizable manifold is rigid iff $b_1(X)=0$ iff Alb$(X)=\{e\}$ iff
$H^1(X,\mathcal O)=0$. | math |
115 | Flat Vector Bundles over Parallelizable Manifolds | math.AG | We study flat vector bundles over complex parallelizable manifolds. | math |
116 | Pieri-type formulas for maximal isotropic Grassmannains via triple intersections | math.AG | We give an elementary proof of the Pieri-type formula in the cohomology of a
Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic
vector space. This proof proceeds by explicitly computing a triple intersection
of Schubert varieties. The decisive step is an exact description of the
intersection of two Schubert varieties, from which the multiplicities (which
are powers of 2) in the Pieri-type formula are immediately obvious. | math |
117 | On quantum cohomology rings of partial flag varieties | math.AG | The main goal of this paper is to give a unified description for the
structure of the small quantum cohomology rings for all homogeneous spaces of
SL_n(C). | math |
118 | A conjectural description of the tautological ring of the moduli space of curves | math.AG | The purpose of this paper is to formulate a number of conjectures giving a
rather complete description of the tautological ring of M_g and to discuss the
evidence for these conjectures. | math |
119 | A non-vanishing result for the tautological ring of {\cal M}_g | math.AG | Looijenga recently proved that the tautological ring of M_g vanishes in
degree d>g-2 and is at most one-dimensional in degree g-2, generated by the
class of the hyperelliptic locus. Here we show that K_{g-2} is non-zero on M_g.
The proof uses the Witten conjecture, proven by Kontsevich. With similar
methods, we expect to be able to prove some (possibly all) of the identities in
degree g-2 in the tautological ring that are part of the author's conjectural
explicit description of the ring. | math |
120 | Intersection Numbers on the Moduli Spaces of Stable Maps in Genus 0 | math.AG | Let $V$ be a smooth, projective, convex variety. We define tautological
$\psi$ and $\kappa$ classes on the moduli space of stable maps $\M_{0,n}(V)$,
give a (graphical) presentation for these classes in terms of boundary strata,
derive differential equations for the generating functions of the Gromov-Witten
invariants of $V$ twisted by these tautological classes, and prove that these
intersection numbers are completely determined by the Gromov-Witten invariants
of $V$. This results in families of Frobenius manifold structures on the
cohomology ring of $V$ which includes the quantum cohomology as a special case. | math |
121 | Germs of de Rham cohomology classes which vanish at the generic point | math.AG | We show that hypergeometric differential equations, unitary and Gauss-Manin
connections give rise to de Rham cohomology sheaves which do not admit a
Bloch-Ogus resolution. The latter is in contrast to Panin's theorem asserting
that corresponding \'etale cohomology sheaves do fulfill Bloch-Ogus theory. | math |
122 | K. Saito's Duality for Regular Weight Systems and Duality for Orbifoldized Poincare Polynomials | math.AG | We will show that the duality for regular weight systems introduced by K.
Saito can be interpreted as the duality for orbifoldized Poincare polynomials. | math |
123 | Characteristic varieties of algebraic curves | math.AG | We study tori attached to the fundamental groups of plane curves with
arbitrary singularities. These tori provide complete information about homology
of finite abelian covers of the plane branched along the curve. We calculate
these tori in terms of certain linear systems determined by the singularities
of the curve. In the case of the complements to a union of lines they can be
calculated from the lattice of the arrangement and are closely related to the
components of the space of Aomoto complexes with prescribed homology. | math |
124 | Lectures on Exotic Algebraic Structures on Affine Spaces | math.AG | These notes are based on the lecture courses given at the
Ruhr-Universit{\"a}t-Bochum (03--08.02.1997) and at the Universit{\'e} Paul
Sabatier (Toulouse, 08-12.01.1996). | math |
125 | Affine modifications and affine hypersurfaces with a very transitive automorphism group | math.AG | We study a kind of modification of an affine domain which produces another
affine domain. First appeared in passing in the basic paper of O. Zariski
(1942), it was further considered by E.D. Davis (1967). The first named author
applied its geometric counterpart to construct contractible smooth affine
varieties non-isomorphic to Euclidean spaces. Here we provide certain
conditions which guarantee preservation of the topology under a modification.
As an application, we show that the group of biregular automorphisms of the
affine hypersurface $X \subset C^{k+2}$ given by the equation
$uv=p(x_1,...,x_k)$ where $p \in C[x_1,...,x_k],$ acts $m-$transitively on the
smooth part reg$X$ of $X$ for any $m \in N.$ We present examples of such
hypersurfaces diffeomorphic to Euclidean spaces. | math |
126 | The local monodromy as a generalized algebraic correspondence | math.AG | In the paper we show that for a normal-crossings degeneration $Z$ over the
ring of integers of a local field with $X$ as generic fibre, the local
monodromy operator and its powers determine invariant cocycle classes under the
decomposition group in the cohomology of the product $X \times X$. More
precisely, they also define algebraic cycles on the special fibre of a
resolution of $Z \times Z$. In the paper, we give an explicit description of
these cycles for a degeneration with at worst triple points as singularities.
These cycles explain geometrically the presence of poles on specific local
factors of the L-function related to $X \times X$. | math |
127 | Algebraic theory of characteristic classes of bundles with connection | math.AG | This is a survey on the topic explained in the title, for the proceedings on
the K-theory 1997 summer institute in Seattle. | math |
128 | The Pfaffian Calabi-Yau, its Mirror, and their link to the Grassmannian G(2,7) | math.AG | The rank 4 locus of a general skew-symmetric 7x7 matrix gives the pfaffian
variety in P^20 which is not defined as a complete intersection. Intersecting
this with a general P^6 gives a Calabi-Yau manifold. An orbifold construction
seems to give the 1-parameter mirror-family of this. However, corresponding to
two points in the 1-parameter family of complex structures, both with maximally
unipotent monodromy, are two different mirror-maps: one corresponding to the
general pfaffian section, the other to a general intersection of G(2,7) in P^20
with a P^13. Apparently, the pfaffian and G(2,7) sections constitute different
parts of the A-model (Kahler structure related) moduli space, and, thus,
represent different parts of the same conformal field theory moduli space. | math |
129 | On the zeta-function of a polynomial at infinity | math.AG | We use the notion of Milnor fibres of the germ of a meromorphic function and
the method of partial resolutions for a study of topology of a polynomial map
at infinity (mainly for calculation of the zeta-function of a monodromy). It
gives effective methods of computation of the zeta-function for a number of
cases and a criterium for a value to be atypical at infinity. | math |
130 | Characteristic Classes of Hypersurfaces and Characteristic Cycles | math.AG | We give a new formula for the Chern-Schwartz-MacPherson class of a
hypersurface in a nonsigular compact complex analytic variety. In particular
this formula generalizes our previous result on the Euler characteristic of
such a hypersurface. Two different approaches are presented. The first is based
on the theory of characteristic cycle and the works of Sabbah,
Briancon-Maisonobe-Merle, and Le-Mebkhout. In particular, this approach leads
to a simple proof of a formula of Aluffi for the above mentioned class. The
second approach uses Verdier's specialization property of the
Chern-Schwartz-MacPherson classes. Some related new formulas are also given. | math |
131 | On maximal curves having classical Weierstrass gaps | math.AG | We study geometrical properties of maximal curves having classical
Weierstrass gaps. | math |
132 | Preuve d'une conjecture de Frenkel-Gaitsgory-Kazhdan-Vilonen | math.AG | We prove a conjecture of Frenkel-Gaitsgory-Kazhdan-Vilonen on some
exponential sums related to the geometric Langlands correspondence. Our main
ingredients are the resolution of Lusztig scheme of lattices introduced by
Laumon and the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber. | math |
133 | Varieties of sums of powers | math.AG | The variety of sums of powers of a homogeneous polynomial of degree d in n
variables is defined and investigated in some examples, old and new. These
varieties are studied via apolarity and syzygies. Classical results of
Sylvester (1851), Hilbert (1888), Dixon and Stuart (1906) and some more recent
results of Mukai (1992) are presented together with new results for the cases
(n,d)=(3,8), (4,2), (5,3). In the last case the variety of sums of 8 powers of
a general cubic form is a Fano 5-fold of index 1 and degree 660. | math |
134 | A Base Point Free Theorem of Reid Type, II | math.AG | Let $X$ be a complete algebraic variety over {\bf C}. We consider a log
variety $(X,\Delta)$ that is weakly Kawamata log terminal. We assume that
$K_X+\Delta$ is a {\bf Q}-Cartier {\bf Q}-divisor and that every irreducible
component of $\lfloor \Delta \rfloor$ is {\bf Q}-Cartier. A nef and big Cartier
divisor $H$ on $X$ is called {\it nef and log big} on $(X,\Delta)$ if $H |_B$
is nef and big for every center $B$ of non-"Kawamata log terminal"
singularities for $(X,\Delta)$. We prove that, if $L$ is a nef Cartier divisor
such that $aL-(K_X+\Delta)$ is nef and log big on $(X,\Delta)$ for some $a \in$
{\bf N}, then the complete linear system $| mL |$ is base point free for $m \gg
0$. | math |
135 | An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation | math.AG | We give an algorithm to compute the following cohomology groups on $U = \C^n
\setminus V(f)$ for any non-zero polynomial $f \in \Q[x_1, ..., x_n]$; 1.
$H^k(U, \C_U)$, $\C_U$ is the constant sheaf on $U$ with stalk $\C$. 2. $H^k(U,
\Vsc)$, $\Vsc$ is a locally constant sheaf of rank 1 on $U$. We also give
partial results on computation of cohomology groups on $U$ for a locally
constant sheaf of general rank and on computation of $H^k(\C^n \setminus Z,
\C)$ where $Z$ is a general algebraic set. Our algorithm is based on
computations of Gr\"obner bases in the ring of differential operators with
polynomial coefficients. | math |
136 | Degeneration of curves and analytic deformation | math.AG | Let C_1 and C_2 be two degenerations of genus g curves. We prove that if two
degenerations defines the same conjugacy classes in the mapping class group,
they are equivalent under analytic deformations. | math |
137 | Singularities | math.AG | This article recounts the interaction of topology and singularity theory
(mainly singularities of complex algebraic varieties) which started in the
early part of this century and bloomed in the 1960's with the work of
Hirzebruch, Brieskorn, Milnor and others. Some of the topics are followed to
the present day. (A chapter for the book "History of Topology", ed. I. M.
James) | math |
138 | Vector bundles on G(1,4) without intermediate cohomology | math.AG | We characterize the vector bundles on G(1,4) that have no intermediate
cohomology. We obtain them from extensions of the universal bundles and others
related with them. In particular, we get a characterization of the universal
vector bundles from their cohomology. | math |
139 | Non-resonance D-modules over arrangements of hyperplanes | math.AG | The aim of this note is a combinatorial description of a category of
$D$-modules over an affine space, smooth along the stratification defined by an
arrangement of hyperplanes. These $D$-modules are assumed to satisfy certain
non-resonance condition. The main result, see Theorem 4.1, generalizes
[S.Khoroshkin, $D$-modules over arrangements of hyperplanes, Comm. in Alg.
23(9) (1995), 3481-3504]. | math |
140 | Grassmannian of $k((z))$: Picard Group, Equations and Automorphisms | math.AG | This paper aims at generalizing some geometric properties of Grassmannians of
finite dimensional vector spaces to the case of Grassmannnians of infinite
dimensional ones, in particular for that of $k((z))$. It is shown that the
Determinant Line Bundle generates its Picard Group and that the Pl\"ucker
equations define it as closed subscheme of a infinite projective space.
Finally, a characterization of finite dimensional projective spaces in
Grassmannians allows us to offer an approach to the study of the automorphism
group. | math |
141 | Remarks on formal deformations and Batalin-Vilkovisky algebras | math.AG | This note consists of two parts. Part I is an exposition of (a part of) the
V.Drinfeld's letter, [D].
The sheaf of algebras of polyvector fields on a Calabi-Yau manifold, equipped
with the Schouten bracket, admits a structure of a Batalin-Vilkovisky algebra.
This fact was probably first noticed by Z.Ran, [R]. Part II is devoted to some
generalizations of this remark. | math |
142 | On the Severi varieties of surfaces in P^3 | math.AG | The Severi variety V_{n,d} of a smooth projective surface S is defined as the
subvariety of the linear system |O_S(n)|, which parametrizes curves with d
nodes. We show that, for a general surface S of degree k in P^3 and for all
n>k-1, d=0,...,dim(|O_S(n)|), there exists one component of V_{n,d} which is
reduced, of the expected dimension dim(|O_S(n)|)-d. Components of the expected
dimension are the easiest to handle, trying to settle an enumerative geometry
for singular curves on surfaces. On the other hand, we also construct examples
of reducible Severi varieties, on general surfaces of degree k>7 in P^3. | math |
143 | Castelnuovo regularity for smooth projective varieties of dimensions 3 and 4 | math.AG | Castelnuovo-Mumford regularity is an important invariant of projective
algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a
bound for regularity in terms of the codimension and degree. This conjecture is
known to be true for curves (Gruson-Lazarsfeld-Peskine) and smooth surfaces
(Pinkham, Lazarsfeld), but not in general.
The purpose of this paper is to give new bounds for the regularity of smooth
varieties in dimensions 3 and 4 that are only slightly worse than the optimal
ones suggested by the conjecture. Our method yields new bounds up to dimension
14, but as they get progressively worse for higher dimensions, we have not
written them down here. | math |
144 | Castelnuovo-Mumford Regularity of Smoth Threefolds in P^5 | math.AG | Castelnuovo-Mumford regularity is an important invariant of projective
algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a
bound for regularity in terms of the codimension and degree,i.e.,
Castelnuovo-Mumford regularity of a given variety $X$ is less than or equal to
$deg(X)-codim(X)+1$. This regularity conjecture (including classification of
examples on the boundary) was verified for integral curves (Castelnuovo,
Gruson, Lazarsfeld and Peskine), and for smooth surfaces (Pinkham, Lazarsfeld).
In this paper we prove that $reg(X) \le deg(X)-1$ for smooth threefolds $X$ in
P^5 and that the only varieties on the boundary are the Segre threefold and the
complete intersection of two quadrics. Furthermore, every smooth threefold $X$
in P^5 is $k$-normal for all $k \ge deg(X)-4$, which is the optimal bound as
the Palatini 3-fold of degree 7 shows. | math |
145 | Varietes de modules alternatives | math.AG | Let X be a projective irreducible smooth algebraic variety. A "fine moduli
space" of sheaves on X is a family F of coherent sheaves on X parametrized by
an integral variety M such that : F is flat on M; for all distinct points x, y
of M the sheaves F_x, F_y on X are not isomorphic and F is a complete
deformation of F_x; F has an obvious local universal property. We define also a
"fine moduli space defined locally", where F is replaced by a family (F_i),
where F_i is defined on an open subset U_i of M, the U_i covering M. This paper
is devoted to the study of such fine moduli spaces. We first give some general
results, and apply them in three cases on the projective plane : the fine
moduli spaces of prioritary sheaves, the fine moduli spaces consisting of
simple rank 1 sheaves, and those which come from moduli spaces of morphisms. In
the first case we give an exemple of a fine moduli space defined locally but
not globally, in the second an exemple of a maximal non projective fine moduli
space, and in the third we find a projective fine moduli space consisting of
simple torsion free sheaves, containing stable sheaves, but which is different
from the corresponding moduli space of stable sheaves. | math |
146 | Rational Curves on the Space of Determinantal Nets of Conics | math.AG | We describe the Hilbert scheme components parametrizing lines and conics on
the space of determinantal nets of conics, N. As an application, we use the
quantum Lefschetz hyperplane principle to compute the instanton numbers of
rational curves on a complete intersection Calabi-Yau threefold in N. We also
compute the number of lines and conics on some Calabi-Yau sections of
non-decomposable vector bundles on N. The paper contains a brief summary of the
A-model theory leading up to Givental-Kim's quantum Lefschetz hyperplane
principle. | math |
147 | Noncommutative geometry based on commutator expansions | math.AG | We develop an approach to noncommutative algebraic geometry ``in the
perturbative regime" around ordinary commutative geometry. Let R be a
noncommutative algebra and A=R/[R,R] its commutativization. We describe what
should be the formal neighborhood of M=Spec(A) in the (nonexistent) space
Spec(R). This is a ringed space (M,O) where O is a certain sheaf of
noncommutative rings on M. Such ringed spaces can be glued together to form
more global objects called NC-schemes. We are especially interested in
NC-manifolds, NC-schemes for which the completion of O at every point of M is
isomorphic to the algebra of noncommutative power series (completion of the
free associative algebra). An explicit description of the simplest NC-manifold,
the affine space, is given by using the Feynman-Maslov calculus of ordered
operators. We show that many familiar algebraic varieties can be naturally
enlarged to NC-manifolds. Among these are all the classical flag varieties and
all the smooth moduli spaces of vector bundles. | math |
148 | Index 1 covers of log terminal surface sigularities | math.AG | We shall investigate index 1 covers of 2-dimensional log terminal
singularities. The main result is that the index 1 cover is canonical if the
characteristic of the base field is different from 2 or 3. We also give some
counterexamples in the case of characteristic 2 or 3. By using this result, we
correct an error in a previous paper. | math |
149 | Monodromy weight filtration is independent of l | math.AG | In this paper, we prove the l-independence of monodromy weight filtration for
a geometrically smooth variety over an equicharacteristic local field. We also
prove the l-independence for the geometric monodromy representation on the
associated graded module of weight monodromy filtration. | math |
150 | Real Algebraic Threefolds III: Conic Bundles | math.AG | This is the third of a series of papers studying real algebraic threefolds,
but the methods are mostly independent from the previous two. Let $f:X\to S$ be
a map of a smooth projective real algebraic 3-fold to a surface $S$ whose
general fibers are rational curves.
Assume that the set of real points of $X$ is an orientable 3-manifold $M$.
The aim of the paper is to give a topological description of $M$. It is shown
that $M$ is either Seifert fibered or a connected sum of lens spaces. Much
stronger results hold if $S$ is rational. | math |
151 | Primitive Forms, Topological LG models coupled to Gravity and Mirror Symmetry | math.AG | In this paper, we will describe the mathematical foundation of topological
Landau-Ginzburg (LG) models coupled to gravity at genus 0 in terms of primitive
forms. We also discuss the mirror symmetry for Calabi-Yau manifolds and CP^1 in
our context. We will show that the mirror partner of CP^1 is the theory of
primitive form associated to f=z+qz^{-1}. | math |
152 | Equivariant cohomology and equivariant intersection theory | math.AG | This text is an introduction to equivariant cohomology, a classical tool for
topological transformation groups, and to equivariant intersection theory, a
much more recent topic initiated by D. Edidin and W. Graham. It is based on
lectures given at Montr\'eal is Summer 1997.
Our main aim is to obtain explicit descriptions of cohomology or Chow rings
of certain manifolds with group actions which arise in representation theory,
e.g. homogeneous spaces and their compactifications.
As another appplication of equivariant intersection theory, we obtain simple
versions of criteria for smoothness or rational smoothness of Schubert
varieties (due to Kumar, Carrell-Peterson and Arabia) whose statements and
proofs become quite transparent in this framework. | math |
153 | A set on which the Lojasieewicz exponent at infinity is attained | math.AG | We show that for a polynomial mapping F = (f_1,...,f_m): C^n \to C^m the
Lojasiewicz exponent at infinity of F is attained on the set {z \in C^n :
f_1(z)...f_m(z) = 0} | math |
154 | Algebraic monoids and group embeddings | math.AG | We study the geometry of algebraic monoids. We prove that the group of
invertible elements of an irreducible algebraic monoid is an algebraic group,
open in the monoid. Moreover, if this group is reductive, then the monoid is
affine. We then give a combinatorial classification of reductive monoids by
means of the theory of spherical varieties. | math |
155 | On the Moduli space of diffeomorphic algebraic surfaces | math.AG | It is proved that the number of deformation types of complex structures on a
fixed oriented smooth four-manifold can be arbitrarily large. The considered
examples are locally simple abelian covers of rational surfaces. | math |
156 | Boundedness of semistable principal bundles on a curve, with classical semisimple structure groups | math.AG | In characteristic zero, semistable principal bundles on a nonsingular
projective curve with a semisimple structure group form a bounded family, as
shown by Ramanathan in 1970's using the Narasimhan-Seshadri theorem. This was
the first step in his construction of moduli for principal bundles. In this
paper we prove boundedness in finite characteristics (other than characteristic
2), when the structure group is a semisimple, simply connected algebraic group
of classical type. The main ingredient is an analogue of the Mukai-Sakai
theorem (which says that any vector bundle admits a proper subbundle whose
degree is `not too small') in the present situation. | math |
157 | The Chow ring of a classifying space | math.AG | We define the Chow ring of the classifying space of a linear algebraic group.
In all the examples where we can compute it, such as the symmetric groups and
the orthogonal groups, it is isomorphic to a natural quotient of the complex
cobordism ring of the classifying space, a topological invariant. We apply this
to get torsion information on the Chow groups of varieties defined as quotients
by finite groups. This generalizes Atiyah and Hirzebruch's use of such
varieties to give counterexamples to the Hodge conjecture with integer
coefficients. | math |
158 | A note on k-jet ampleness on surfaces | math.AG | We prove Reider type criterions for k-jet spannedness and k-jet ampleness of
adjoint bundles for surfaces with at most rational singularities. Moreover, we
prove that on smooth surfaces [n(n+4)/4]-very ampleness implies n-jet
ampleness. | math |
159 | Pluricanonical systems on surfaces with small K^2 | math.AG | We prove that the bicanonical system on a surface of general type with K^2=4
has no base components and describe clusters contracted by 4K_X for a numerical
Godeaux surface and 3K_X for a numerical Campedelli surface. | math |
160 | On plane maximal curves | math.AG | The genus of a maximal curve over a finite field with r^2 elements is either
g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus
g_0 or g_1 have been characterized up to isomorphism. A natural genus to be
studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism
maximal curves known when r \equiv 3 (mod 4). Here, a maximal curve with genus
g_2 and a non-singular plane model is characterized as a Fermat curve of degree
(r+1)/2. | math |
161 | Mori conic bundles with a reduced log terminal boundary | math.AG | We study the local structure of Mori contractions $f\colon X\to Z$ of
relative dimension one under an additional assumption that there exists a
reduced divisor $S$ such that $K_X+S$ is plt and anti-ample. | math |
162 | Effectivity of Arakelov divisors and the theta divisor of a number field | math.AG | We introduce the notion of an effective Arakelov divisor for a number field
and the arithmetical analogue of the dimension of the space of sections of a
line bundle. We study the analogue of the theta divisor for a number field. | math |
163 | Calculating cohomology groups of moduli spaces of curves via algebraic geometry | math.AG | We compute the first, second, third, and fifth rational cohomology groups of
the moduli space of stable n-pointed genus g curves, for all g and n, using
(mostly) algebro-geometric techniques. | math |
164 | Enumerative geometry of plane curves of low genus | math.AG | We collect various known results (about plane curves and the moduli space of
stable maps) to derive new recursive formulas enumerating low genus plane
curves of any degree with various behaviors. Recursive formulas are given for
the characteristic numbers of rational plane curves, elliptic plane curves, and
elliptic plane curves with fixed complex structure. Recursions are also given
for the number of elliptic (and rational) plane curves with various
"codimension 1" behavior (cuspidal, tacnodal, triple pointed, etc., as well as
the geometric and arithmetic sectional genus of the Severi variety). We compute
the latter numbers for genus 2 and 3 plane curves as well. We rely on results
of Caporaso, Diaz, Getzler, Harris, Ran, and especially Pandharipande. | math |
165 | Cohomological invariants of complex manifolds coming from extremal rays | math.AG | In the present paper Mori extremal rays of a smooth projective manifold X are
divided into two classes: L-supported and L-negligible (where ``L'' stands for
``Lefschetz'' since the division comes from Hard Lefschetz Theorem). Roughly
speaking: L-supported rays are strongly distinguishable in topology while
L-negligible rays have very mild geometry.
Each L-supported ray R defines hyperplane in H^2(X,R) on which Lefschetz
duality degenerates so it is a cohomology ring invariant. The hyperplane
carries a multiplicity (cohomology ring invariant) which is related to the
geometry of the ray R. The number of L-supported rays is bounded.
Although the number of L-negligible rays may be infinite and they are
invisible in the cohomology ring, their geometry is easier than that of
L-supported rays. They are classifieable in low dimensions. Each L-negligible
ray contains lots of ``good'' rational curves whose deformation is of expected
dimension. In effect, L-negligible rays are invariant under deformations of
complex structure and can be used to compute Gromov-Witten invariants in
symplectic geometry. | math |
166 | Singularities of 2theta-divisors in the Jacobian | math.AG | We study several subseries of the space of second order theta functions on
the Jacobian of a non-hyperelliptic curve. In particular, we are interested in
the subseries P\Gamma_{00} consisting of 2theta-divisors having multiplicity at
least 4 at the origin, or, equivalently, containing the surface C-C, and in its
analogues consisting of 2theta-divisors having higher multiplicities at the
origin, containing the four-fold Sym^2C-Sym^2C, or singular along the surface
C-C. We use rank 2 vector bundles with a given number of global sections to
prove canonical isomorphisms between quotients of the above introduced
subseries and vector spaces defined by the canonical divisor. | math |
167 | Massey and Fukaya products on elliptic curves | math.AG | This note is a continuation of our paper with E. Zaslow "Categorical mirror
symmetry: the elliptic curve", math.AG/9801119. We compare some triple Massey
products on elliptic curve with the corresponding Fukaya products on the
symplectic torus and recover the classical identity due to Kronecker. We also
compute some triple Fukaya products such that the corresponding Massey products
are not correctly defined. | math |
168 | Locally symmetric families of curves and jacobians | math.AG | This paper addresses the following question of Oort: "Are there any postive
dimensional locally symmetric subvarieties of the moduli space of abelian
varieties that are contained in the jacobian locus and contain the jacobian of
at least one smooth curve? Some partial results are given. | math |
169 | On curves covered by the Hermitian curve | math.AG | For each proper divisor d of (r^2-r+1), r being a power of a prime, maximal
curves over a finite field with r^2 elements covered by the Hermitian curve of
genus 1/2((r^2-r+1)/d-1) are constructed. | math |
170 | Character sums associated to finite Coxeter groups | math.AG | We prove a character sum identity for Coxeter arrangements which is a finite
field analogue of Macdonald's conjecture proved by Opdam. | math |
171 | Germs of arcs on singular algebraic varieties and motivic integration | math.AG | We study the scheme of formal arcs on a singular algebraic variety and its
images under truncations. We prove a rationality result for the Poincare series
of these images which is an analogue of the rationality of the Poincare series
associated to p-adic points on a p-adic variety. The main tools which are used
are semi-algebraic geometry in spaces of power series and motivic integration
(a notion introduced by M. Kontsevich). In particular we develop the theory of
motivic integration for semi-algebraic sets of formal arcs on singular
algebraic varieties, we prove a change of variable formula for birational
morphisms and we prove a geometric analogue of a result of Oesterle. | math |
172 | Motivic Igusa zeta functions | math.AG | We define motivic analogues of Igusa's local zeta functions. These functions
take their values in a Grothendieck group of Chow motives. They specialize to
p-adic Igusa local zeta functions and to the topological zeta functions we
introduced several years ago. We study their basic properties, such as
functional equations, and their relation with motivic nearby cycles. In
particular the Hodge spectrum of a singular point of a function may be
recovered from the Hodge realization of these zeta functions. | math |
173 | Chiral de Rham complex | math.AG | The aim of this note is to define certain sheaves of vertex algebras on
smooth manifolds. For each smooth complex algebraic (or analytic) manifold $X$,
we construct a sheaf $\Omega^{ch}_X$, called the {\bf chiral de Rham complex}
of $X$. It is a sheaf of vertex algebras in the Zarisky (or classical)
topology, It comes equipped with a $\BZ$-grading by {\it fermionic charge}, and
the {\it chiral de Rham differential} $d_{DR}^{ch}$, which is an endomorphism
of degree 1 such that $(d_{DR}^{ch})^2=0$. One has a canonical embedding of the
usual de Rham complex $(\Omega_X, d_{DR})\hra (\Omega_X^{ch}, d_{DR}^{ch})$
which is a quasiisomorphism.
If $X$ is Calabi-Yau then this sheaf admits an N=2 supersymmetry. For some
$X$ (for example, for curves or for the flag spaces $G/B$), one can construct
also a purely even analogue of this sheaf, a {\it chiral structure sheaf}
$\CO^{ch}_X$. For the projective line, the space of global sections of the last
sheaf is the irreducible vacuum $\hsl(2)$-module on the critical level. | math |
174 | On $-K^2$ for normal surface singularities | math.AG | In this paper we show the lower bound of the set of non-zero $-K^2$ for
normal surface singularities establishing that this set has no accumulation
points from above. We also prove that every accumulation point from below is a
rational number and every positive integer is an accumulation point. Every
rational number can be an accumulation point modulo $\bZ$. We determine all
accumulation points in $[0, 1]$. If we fix the value $-K^2$, then the values of
$p_g$, $p_a$, mult, embdim and the numerical indices are bounded, while the
numbers of the exceptional curves are not bounded. | math |
175 | Motivic exponential integrals and a motivic Thom-Sebastiani Theorem | math.AG | We introduce motivic analogues of p-adic exponential integrals. We prove a
basic multiplicativity property from which we deduce a motivic analogue of the
Thom-Sebastiani Theorem. In particular, we obtain a new proof of the
Thom-Sebastiani Theorem for the Hodge spectrum of (non isolated) singularities
of functions. | math |
176 | Elliptic Gromov - Witten invariants and the generalized mirror conjecture | math.AG | A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic
terms of semi-simple Frobenius structures and complex oscillating integrals is
formulated. The proof of the conjecture is given for torus-equivariant Gromov -
Witten invariants of compact K\"ahler manifolds with isolated fixed points and
for concave bundle spaces over such manifolds. Several results on genus 0
Gromov - Witten theory include: a non-linear Serre duality theorem, its
application to the genus 0 mirror conjecture, a mirror theorem for concave
bundle spaces over toric manifolds generalizing a recent result of B. Lian, K.
Liu and S.-T. Yau. We also establish a correspondence (see the extensive
footnote in section 4) between their new proof of the genus 0 mirror conjecture
for quintic 3-folds and our proof of the same conjecture given two years ago. | math |
177 | Intersection pairing for arithmetic cycles with degenerate Green currents | math.AG | In this note, we would like to propose a suitable extension of the arithmetic
Chow group of codimension one, in which the Hodge index theorem holds. We also
prove an arithmetic analogue of Bogomolov's instability theorem for rank 2
vector bundles on arbitrary regular projective arithmetic varieties. | math |
178 | Ray class fields of global function fields with many rational places | math.AG | A general type of ray class fields of global function fields is investigated.
The systematic computation of their genera leads to new examples of curves over
finite fields with comparatively many rational points. | math |
179 | Strange duality and polar duality | math.AG | We describe a relation between Arnold's strange duality and a polar duality
between the Newton polytopes which is mostly due to M.~Kobayashi. We show that
this relation continues to hold for the extension of Arnold's strange duality
found by C.~T.~C.~Wall and the author. By a method of Ehlers-Varchenko, the
characteristic polynomial of the monodromy of a hypersurface singularity can be
computed from the Newton diagram. We generalize this method to the isolated
complete intersection singularities embraced in the extended duality. We use
this to explain the duality of characteristic polynomials of the monodromy
discovered by K.~Saito for Arnold's original strange duality and extended by
the author to the other cases. | math |
180 | Poisson structures on algebraic threefolds | math.AG | The aim of this paper is to find all algebraic threefolds admitting
quasi-regular Poisson structure. There are three types of such varieties:
abelian varieties, smooth flat conic bundles over abelian surfaces and
quotients of the product of a curve and a symplectic surface by a finite group. | math |
181 | Riemann-Roch Theorems for Deligne-Mumford Stacks | math.AG | The goal of this paper is to prove Riemann-Roch type theorems for
Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with
coefficients in representations" and a Chern character, and we prove a
Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it
defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the
Euler characteristic of a coherent sheaf, and some formulas for the different
topological Euler characteristics of complex algebraic stacks. | math |
182 | Chern Classes of Tautological Sheaves on Hilbert Schemes | math.AG | We give an algorithmic description of the action of the Chern classes of
tautological bundles on the cohomology of Hilbert schemes of points on surfaces
within the framework of Nakajima's oscillator algebra. This leads to an
identification of the cohomology ring of Hilbert schemes of the affine plane
with a ring of differential operators on a Fock space. We end with the
computation of the top Segre classes of tautological bundles associated to line
bundles on Hilb^n up to n=7, and give a conjecture for the generating series. | math |
183 | On a series of Gorenstein cyclic quotient singularities admitting a unique projective crepant resolution | math.AG | In this paper we prove that the Gorenstein cyclic quotient singularities of
type \frac 1l (1,..., 1,l-(r-1)) with $l\geq r\geq 2$, have a
\textit{unique}torus-equivariant projective, crepant, partial resolution, which
is ``full'' iff either $l\equiv 0$ mod $% (r-1) $ or $l\equiv 1$ mod $(r-1) $.
As it turns out, if one of these two conditions is fulfilled, then the
exceptional locus of the full desingularization consists of $\lfloor
\frac{l}{r-1} \rfloor$ prime divisors, $\lfloor \frac{l}{r-1}\rfloor - 1$ of
which are isomorphic to the total spaces of $\Bbb{P}_{\Bbb{C}}^1$-bundles over
$\Bbb{P}_{\Bbb{C}%}^{r-2}$. Moreover, it is shown that intersection numbers are
computable explicitly and that the resolution morphism can be viewed as a
composite of successive (normalized) blow-ups. Obviously, the monoparametrized
singularity-series of the above type contains (as its ``first member'') the
well-known Gorenstein singularity defined by the origin of the affine cone
which lies over the $r$-tuple Veronese embedding of $\Bbb{P}_{\Bbb{C}}^{r-1}$. | math |
184 | On crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities | math.AG | An immediate generalization of the classical McKay correspondence for
Gorenstein quotient spaces $\Bbb{C}^{r}/G$ in dimensions $r\geq 4$ would
primarily demand the existence of projective, crepant, full desingularizations.
Since this is not always possible, it is natural to ask about special classes
of such quotient spaces which would satisfy the above property. In this paper
we give explicit necessary and sufficient conditions under which 2-parameter
series of Gorenstein cyclic quotient singularities have torus-equivariant
resolutions of this specific sort in all dimensions. | math |
185 | On a conjecture of Le Bruyn | math.AG | Given a generic field extension F/k of degree n>3 (i.e. the Galois group of
the normal closure of F is isomorphic to the symmetric group $S_n$), we prove
that the norm torus, defined as the kernel of the norm map
$N:R_{F/k}(G_m)\to\G_m$, is not rational over k. | math |
186 | Mirror Symmetry and Toric Degenerations of Partial Flag Manifolds | math.AG | In this paper we propose and discuss a mirror construction for complete
intersections in partial flag manifolds $F(n_1, ..., n_l, n)$. This
construction includes our previous mirror construction for complete
intersection in Grassmannians and the mirror construction of Givental for
complete flag manifolds. The key idea of our construction is a degeneration of
$F(n_1, ..., n_l, n)$ to a certain Gorenstein toric Fano variety $P(n_1, ...,
n_l, n)$ which has been investigated by Gonciulea and Lakshmibai. We describe a
natural small crepant desingularization of $P(n_1, ..., n_l, n)$ and prove a
generalized version of a conjecture of Gonciulea and Lakshmibai on the singular
locus of $P(n_1, ..., n_l, n)$. | math |
187 | Triades et familles de courbes gauches | math.AG | Let $A$ be a noetherian ring and $R_A$ be the graded ring $A[X,Y,Z,T]$. In
this article we introduce the notion of a triad, which is a generalization to
families of curves in ${\bf P}^3_A$ of the notion of Rao module. A triad is a
complex of graded $R_A$-modules $(L_1 \to L_0 \to L_{-1})$ with certain
finiteness hypotheses on its cohomology modules.
A pseudo-isomorphism between two triads is a morphism of complexes which
induces an isomorphism on the functors $ h_0 (L\otimes .)$ and a monomorphism
on the functors $h_{-1} (L\otimes .)$. One says that two triads are
pseudo-isomorphic if they are connected by a chain of pseudo-isomorphisms.
We show that to each family of curves is associated a triad, unique up to
pseudo-isomorphism, and we show that the map $\{\hbox{families of curves}\}\to
\{\hbox{triads}\}$ has almost all the good properties of the map
$\{\hbox{curves}\}\to \{\hbox{Rao modules}\}$.
In a section of examples, we show how to construct triads and families of
curves systematically starting from a graded module and a sub-quotient (that is
a submodule of a quotient module), and we apply these results to show the
connectedness of $H_{4,0}$. | math |
188 | Albanese and Picard 1-motives | math.AG | We define, in a purely algebraic way, 1-motives $Alb^{+}(X)$, $Alb^{-}(X)$,
$Pic^{+}(X)$ and $Pic^{-}(X)$ associated with any algebraic scheme $X$ over an
algebraically closed field of characteristic zero. For $X$ over $\C$ of
dimension $n$ the Hodge realizations are, respectively, $H^{2n-1}(X)(n)$,
$H_{1}(X)$, $H^{1}(X)(1)$ and $H_{2n-1}(X)(1-n)$. | math |
189 | Chern classes and the periods of mirrors | math.AG | We show how Chern classes of a Calabi Yau hypersurface in a toric Fano
manifold can be expressed in terms of the holomorphic at a maximal degeneracy
point period of its mirror. We also consider the relation between Chern classes
and the periods of mirrors for complete intersections in Grassmanian Gr(2,5). | math |
190 | A note on the symplectic structure on the space of G-monopoles | math.AG | Let $G$ be a semisimple complex Lie group with a Borel subgroup $B$. Let
$X=G/B$ be the flag manifold of $G$. Let $C=P^1\ni\infty$ be the projective
line. Let $\alpha\in H_2(X,{\Bbb Z})$. The moduli space of $G$-monopoles of
topological charge $\alpha$ (see e.g. [Jarvis]) is naturally identified with
the space $M_b(X,\alpha)$ of based maps from $(C,\infty)$ to $(X,B)$ of degree
$\alpha$. The moduli space of $G$-monopoles carries a natural hyperk\"ahler
structure, and hence a holomorphic symplectic structure. We propose a simple
explicit formula for the symplectic structure on $M_b(X,\alpha)$. It
generalizes the well known formula for $G=SL_2$ (see e.g. [Atiyah-Hitchin]).
Let $P\supset B$ be a parabolic subgroup. The construction of the Poisson
structure on $M_b(X,\alpha)$ generalizes verbatim to the space of based maps
$M=M_b(G/P,\beta)$. In most cases the corresponding map $T^*M\to TM$ is not an
isomorphism, i.e. $M$ splits into nontrivial symplectic leaves. These leaves
are explicilty described. | math |
191 | A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension | math.AG | The recent two proofs for the (weak) factorization theorem for birational
maps, one by W{\l}odarczyk and the other by
Abramovich-Karu-Matsuki-W{\l}odarczyk rely on the results of Morelli. The
former uses the process for $\pi$-desingularization (the most subtle part of
Morelli's combinatorial algorithm), while the latter uses the strong
factorization of toric (toroidal) birational maps directly. This paper provides
a coherent account of Morelli's work (and its toroidal extension) clarifying
some discrepancies in the original argument. | math |
192 | Jacobian Conjecture and Nilpotent Mappings | math.AG | We prove the equivalence of the Jacobian Conjecture (JC(n)) and the
Conjecture on the cardinality of the set of fixed points of a polynomial
nilpotent mapping (JN(n)) and prove a series of assertions confirming JN(n). | math |
193 | On a Chisini Conjecture | math.AG | Chisini's conjecture asserts that for a cuspidal curve $B\subset \mathbb P^2$
a generic morphism $f$ of a smooth projective surface onto $\mathbb P^2$ of
degree $\geq 5$, branched along $B$, is unique up to isomorphism. We prove that
if $\deg f$ is greater than the value of some function depending on the degree,
genus, and number of cusps of $B$, then the Chisini conjecture holds for $B$.
This inequality holds for many different generic morphisms. In particular, it
holds for a generic morphism given by a linear subsystem of the $m$th canonical
class for almost all surfaces with ample canonical class. | math |
194 | 3-fold log flips according to V.V.Shokurov | math.AG | In this paper, I review the section 8 of V.V.Shokurov's paper '3-fold log
flips'. | math |
195 | On the minimal free resolution of non-special curves in P^3 | math.AG | Here we prove that the minimal free resolution of a general space curve of
large degree (e.g. a general space curve of degree d and genus g with d g+3,
except for finitely many pairs (d,g)) is the expected one. A similar result
holds even for general curves with special hyperplane section and, roughly, d
g/2. The proof uses the so-called methode d'Horace. | math |
196 | Chow quotients and projective bundle formulas for Euler-Chow series | math.AG | Given a projective algebraic variety $X$, let $\Pi_p(X)$ denote the monoid of
effective algebraic equivalence classes of effective algebraic cycles on $X$.
The $p$-th Euler-Chow series of $X$ is an element in the formal monoid-ring
$Z[[\Pi_p(X)]]$ defined in terms of Euler characteristics of the Chow varieties
$\cvpd{p}{\alpha}{X}$ of $X$, with $\alpha \in\Pi_p(X)$. We provide a
systematic treatment of such series, and give projective bundle formulas which
generalize previous results by B. Lawson and S.S.Yau and Elizondo. The
techniques used involve the Chow quotients introduced by Kapranov, and this
allows the computation of various examples including some Grassmannians and
flag varieties. There are relations between these examples and representation
theory, and further results point to interesting connections between Euler-Chow
series for certain varieties and the topology of the closure of moduli spaces
$M_{0,n+1}$. | math |
197 | Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye | math.AG | We give a geometric interpretation of the base change homomorphism between
the Hecke algebra of GL(n) for an unramified extension of local fields of
positive characteristic. For this, we use some results of Ginzburg, Mirkovic
and Vilonen related to the geometric Satake isomorphism. We give new proof for
these results in the positive characteristic case.
By using that geometric interpretation of the base change homomorphism, we
prove the fundamental lemma of Jacquet and Ye for arbitrary Hecke function in
the the equal characteristic case. | math |
198 | Linear Systems of Plane Curves with Base Points of Equal Multiplicity | math.AG | In this article we address the problem of computing the dimension of the
space of plane curves of degree $d$ with $n$ general points of multiplicity
$m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$,
the dimension is equal to the expected dimension given by the Riemann-Roch
Theorem. Also, systems for which the dimension is larger than expected should
have a fixed part containing a multiple $(-1)$-curve. We reformulate this
conjecture by explicitly listing those systems which have unexpected dimension.
Then we use a degeneration technique developed in a previous article
("Degenerations of Planar Linear Systems", alg-geom/9702015) to show that the
conjecture holds for all $m \leq 12$. | math |
199 | Vector bundles on Fano 3-folds without intermediate cohomology | math.AG | We study the vector bundles without intermediate cohomology on Fano
threefolds of index two, degree d=3,4,5 and Betti number one. We obtain a
complete characterization in the case of rank-two vector bundles. For arbitrary
rank, we give all possible Chern classes of such vector bundles, under some
general condition. | math |