title:
Explicit Enumerative Geometry for the Real Grassmannian of Lines in
  Projective Space

abstract:
  We extend the classical Schubert calculus of enumerative geometry for the
Grassmann variety of lines in projective space from the complex realm to the
real. Specifically, given any collection of Schubert conditions on lines in
projective space which generically determine a finite number of lines, we show
there exist real generic conditions determining the expected number of real
lines. Our main tool is an explicit description of rational equivalences which
also constitutes a novel determination of the Chow rings of these Grassmann
varieties.


title: Mixed Hodge structures of configuration spaces

abstract:
  The symmetric group S_n acts freely on the configuration space of n distinct
points in a quasi-projective variety. In this paper, we study the induced
action of the symmetric group S_n on the de Rham cohomology of this space,
using mixed Hodge theory, combined with methods from the theory of symmetric
functions. (We prove a motivic version of this as well.) As an application of
our results, we calculate the S_n-equivariant Hodge polynomial of the
Fulton-MacPherson compactification X[n] of the configuration space.


title:
GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau
  Hypersurfaces

abstract:
  We present a detailed study of the generalized hypergeometric system
introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in
the context of toric geometry. GKZ systems arise naturally in the moduli theory
of Calabi-Yau toric varieties, and play an important role in applications of
the mirror symmetry. We find that the Gr\"obner basis for the so-called toric
ideal determines a finite set of differential operators for the local solutions
of the GKZ system. At the special point called the large radius limit, we find
a close relationship between the principal parts of the operators in the GKZ
system and the intersection ring of a toric variety. As applications, we
analyze general three dimensional hypersurfaces of Fermat and non-Fermat types
with Hodge numbers up to $h^{1,1}=3$. We also find and analyze several non
Landau-Ginzburg models which are related to singular models.


title:
Constructing Indecomposable Motivic Cohomology Classes on Algebraic
  Surfaces

abstract:
  We describe a method to construct indecomposable classes in Bloch's higher
Chow group $CH^2(X,1)$ on algebraic surfaces over the complex numbers via
transcendental methods and apply it to obtain examples on K3-surfaces and some
surfaces of general type.


title: Geography of Brill-Noether loci for small slopes

abstract:
  Let $X$ be a non-singular projective curve of genus $g\ge2$ over an
algebraically closed field of characteristic zero. Let $\mo$ denote the moduli
space of stable bundles of rank $n$ and degree $d$ on $X$ and $\wn $ the
Brill-Noether loci in $\mo .$ We prove that, if $0\leq d \leq n $ and $\wn $ is
non-empty, then it is irreducible of the expected dimension and smooth outside
$\wnn$. We prove further that in this range $\wn$ is non-empty if and only if
$d>0$, $n\leq d+(n-k)g$ and $(n,d,k) \not= (n,n,n)$. We also prove
irreducibility and non-emptiness for the semistable Brill-Noether loci.


title: Faisceaux automorphes pour GL(n): la premiere construction de Drinfeld

abstract:
  Drinfeld a associe a tout systeme local irreductible de rang 2 sur une
surface de Riemann compacte un "faisceau automorphe". Ce faisceau automorphe
vit sur l'espace de modules des fibres vectoriels de rang 2 sur cette meme
surface de Riemann.
  En fait, Drinfeld a donne deux constructions de ce faisceau automorphe. La
premiere est basee sur l'existence d'un modele de Whittaker pour toute
representation automorphe cuspidale pour GL(2). Elle garde un sens en
caracteristique positive et elle s'etend conjecturalement en rang n arbitraire.
  La seconde construction est basee sur un calcul explicite de l'anneau des
sections globales d'un certain faisceau d'operateurs differentiels sur l'espace
de modules des fibres vectoriels. Elle n'a pour l'instant de sens qu'en
caracteristique nulle mais, par contre, elle est de portee plus generale que la
premiere construction car elle a un sens non seulement pour GL(n) mais aussi
pour un groupe reductif arbitraire.
  Ceci est le texte de deux exposes consacres a la premiere construction de
Drinfeld et a son extension conjecturale en rang n arbitraire, que j'ai donnes
a Luminy lors du Colloque "Fibres vectoriels sur les courbes et theorie de
Langlands", en juin 1995.


title: How to Construct Curves over Finite Fields With Many Points

abstract:
  In this paper we give several methods to construct curves over finite fields
with many points and illustrate this with examples of the results.


title: On the intersections of rational curves with cubic plane curves

abstract:
  Let C be a smooth cubic curve in the complex projective plane. We show that
for every positive integer k, there are only finite number of rational curves
of degree k each intersects the cubic C at exactly one point. The number of
such rational curves is known when k is 1, as a smooth cubic has 9 flexes
points. This number seems to be closely related to the number of plane rational
curves of degree k passing through 3k-1 general points, which has been
computed.


title: Perverse sheaves on a Loop group and Langlands' duality

abstract:
  An intrinsic construction of the tensor category of finite dimensional
representations of the Langlands dual group of G in terms of a tensor category
of perverse sheaves on the loop group, LG, is given. The construction is
applied to the study of the topology of the affine Grassmannian of G and to
establishing a Langlands type correspondence for "automorphic" sheaves on the
moduli space of G-bundles.


title: Evolutions, Symbolic Squares, and Fitting Ideals

abstract:
  Given a reduced local algebra $T$ over a suitable ring or field $k$ we study
the question of whether there are nontrivial algebra surjections $R\to T$ which
induce isomorphisms $\Omega_{R/k}\otimes T \to \Omega_{T/k}$. Such maps, called
evolutions, arise naturally in the study of Hecke algebras, as they implicitly
do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the
existence of non-trivial evolutions of an algebra $T$ can be characterized in
terms of the symbolic square of an ideal defining $T$. We give a
characterization of the symbolic square in terms of Fitting ideals. Using this
and other techniques we show that certain classes of reduced algebras --
codimension 2 Cohen-Macaulay, Codimension 3 Gorenstein, licci algebras in
general, and some others -- admit no nontrivial evolutions. On the other hand
we give examples showing that non-trivial evolutions of reduced Cohen-Macaulay
algebras of codimension 3 do exist in every positive characteristic.


title: Cohomology of compact hyperkaehler manifolds and its applications

abstract:
  This article contains a compression of results from alg-geom/9501001, with
most proofs omitted. We prove that every two points of the connected moduli
space of holomorphically symplectic manifolds can be connected with so-called
``twistor lines'' -- projective lines holomorphically embedded to the moduli
space and corresponding to the hyperk\"ahler structures. This has interesting
implications for the geometry of compact hyperk\"ahler manifolds and of
holomorphic vector bundles over such manifolds.


title: On discriminants of polylinear forms

abstract:
  In this paper we propose a conseptual framework for the observed properties
of discriminants of polylinear forms. The connection with classical problems of
linear algebra is shown. A new class of algebraic varieties (hypergrassmanians)
is introduced, the particular case of which are grassman manifolds. An
algorithm is given for computing the discriminants of polylinear forms of
"boundary format" (hypergrassmanian analogues of plukker coordinates). The
algorithm for computing the discriminants of polylinear forms of general
formats is outlined.


title: The 2-dimensional jacobian conjecture via Klein's program

abstract: We investigate the 2-dimensional jacobian conjecture via Klein's program.


title: Picard group of the moduli spaces of G-bundles

abstract:
  We prove that the Picard group of the moduli space of semistable G-bundles on
any irreducible smooth projective curve is isomorphic with the group of
integers.


title: Normal presentation on Elliptic Ruled surfaces

abstract:
  In this article we determine exactly which line bundles on elliptic ruled
surface X are normally presented. In particular we see that numerical classes
of normally presented divisors form a convex set. (recall that Num(X) is
generated by the class of a minimal section C_0 and by the class of a fiber f
and that C_0 is ample.) As a corollary of the above result we show that Mukai's
conjecture is true for the normal presentation of the it adjoint linear series
for an elliptic ruled surface. In section 5 of this article, we show that if L
is normally presented on X then the homogeneous coordinate ring associated to L
is Koszul. We also give a new proof of the following result due to Butler: if
deg(L) \geq 2g+2 on a curve X of genus g, then L embeds X with Koszul
homogeneous coordinate ring.


title: A few general points of the Hilbert scheme of K3 surfaces

abstract:
  A ribbon D over a variety C is a scheme such that D_{red} = C, the ideal I in
O_D of Cis a line bundle on C and I^2 = 0. A two dimensional ribbon is called a
carpet. In this article we show that if D is a K3 carpet, that is a ribbon
associated to a rational normal scroll with numerical invariants that of a K3
surface,then it can be smoothed to a K3 surface. We also show that not all K3
carpets are smooth points of the Hilbert scheme unlike in the dimension one
case as shown by Bayer and Eisenbud. We prove that only those K3 carpets which
are supported on "balanced" scrolls are smooth points of the Hilbert scheme.


title:
Subadjunction of log canonical divisors for a subvariety of codimension
  2

abstract:
  We obtain a formula which relates the log canonical divisor of the ambient
space with that of a subvariety of codimension 2 by using Knudsen's moduli
space of pointed stable curves of genus 0.


title:
Exceptional and Rigid Sheaves on Surfaces with Anticanonical Class
  without Base Components

abstract:
  The paper consists of three parts. In the first of them different kinds
stability are discussed. In particular, the stability concept with respect to
nef divisor is introduced. A structure of rigid and superrigid vector bundles
on smooth projective surfaces with nef anticanonical class is studied in the
second part. We prove that any superrigid bundle has a unique exceptional
filtration. In the last part we give constructive description of exceptional
bundles on these surfaces.


title: On the structure of semistable rigid sheaves on algebraic surfaces

abstract:
  Let S be a smooth projective surface, K be the canonical class of S and H be
an ample divisor such that H.K<0 . In this paper we prove that for any rigid
(Ext^1(F,F)=0) semistable sheaf F in the sense of Mumford--Takemoto stability
w.r.t. H there exists an exceptional collection (E_1,...,E_n) of sheaves on S
such that F can be constructed from {E_i} by a finite number of extensions.


title:
Symplectic biextensions and a generalization of the Fourier-Mukai
  transform

abstract:
  A generalization of the Fourier-Mukai transform is proposed. The construction
is based on analogy with the classical picture of representations of the
Heisenberg group.


title:
Construction of rank two bundles on ${\bf P}^4$ in positive
  characteristic

abstract:
  Many examples of rank two bundles on ${\bf P}^4$ are constructed in positive
characteristic. Construction depends on constructing certain special bundles on
${\bf P}^3$ which is shown to be equivalent to constructing bundles on ${\bf
P}^4$ (in any characteristic). In fact this is true for projective spaces of
any dimension.


title: Infinitesimal presentations of the Torelli groups

abstract:
  This is a significant revision of the early version of this paper which was
posted last December. The speculative section has been removed in light of some
recent results of Morita and Kawazumi. Numerous typos have been fixed. The
companion paper "The Hodge de Rham Theory of Relative Malcev Completion" has
just been posted.


title: Primitive Calabi-Yau Threefolds

abstract:
  A primitive Calabi-Yau threefold is a non-singular Calabi-Yau threefold which
cannot be written as a crepant resolution of a singular fibre of a degeneration
of Calabi-Yau threefolds. These should be thought as the most basic Calabi-Yau
manifolds; all others should arise through degenerations of these. This paper
first continues the study of smoothability of Calabi-Yau threefolds with
canonical singularities begun in the author's previous paper, ``Deforming
Calabi-Yau Threefolds.'' (alg-geom 9506022). If X' is a non-singular Calabi-Yau
threefold, and f:X'->X is a contraction of a divisor to a curve, we obtain
results on when X is smoothable. We then discuss applications of this result to
the classification of primitive Calabi-Yau threefolds. Combining the
deformation theoretic results of this paper with those of ``Deforming
Calabi-Yau Threefolds,'' we obtain strong restrictions on the class of
primitive Calabi-Yau threefolds. We also speculate on how such work might yield
results on connecting together all moduli spaces of Calabi-Yau threefolds.


title: Quot schemes for flags and Gromov invariants for flag varieties

abstract:
  Using Quot schemes and a localization theorem we study Gromov-Witten
invariants for partial flag varieties. The strategy is to extend A. Bertram's
result of Gromov-Witten invariants for special Schubert varieties of
Grassmannians to the case of partial flag varieties. To do so a Grothendieck's
Quot scheme is generalized for flags and proven to be an irreducible, rational,
smooth, projective variety following Str\o mme \cite{St}.


title: Varieties of CM-type

abstract:
  We introduce the notion of a variety (or more generally a motive) of CM-type
which generalises the well known notion of abelian variety of CM-type. Just as
in that particular case it will turn out that the cohomology of the variety is
determined by purely combinatorial data; the type of the variety. As
applications we will show that the \l-adic representations are given by
algebraic Hecke characters whose algebraic parts are determined by the type and
give a method for computing the discriminant of the N\'eron-Severi group of
super-singular Fermat surfaces.


title: Higher Syzygies of Elliptic Ruled Surfaces

abstract:
  Let L be a normally generated line bundle on X; we say L satisfies property
N_p (notation after Mark Green) if the matrices in the free resolution of R
(the homogeneous coordinate ring of X) over S (the homogeneous coordinate ring
of the projective space corresponding to the complete linear series |L|) have
linear entries until the p-th stage. In this article we prove the following
result: Let X be an elliptic ruled surface and let L be a product of p+1 base
point free and ample line bundles on X. Then L satisfies property N_p. In
particular we prove that numerical classes of all divisors which satisfies
property N_p form a convex set. (Recall that Num(X) is generated by the class
of a minimal section C_0 and by the class of a fiber f and that C_0 is ample.)
As a corollary of the above result we show that the adjoint bundle K_X+(2p+3)A
satisfies property N_p, if A is an ample line bundle.


title:
On non-strict notions of $n$-category and $n$-groupoid via
  multisimplicial sets

abstract:
  In this paper we first give a simplicial approach to the definition of a non
strict $n$-category that we call an $n$-nerve following the idea that a
category could be interpreted as a simplicial set, and we prove that our
construction generalises the case of the usual non strict 2-category. Next we
give a simplicial definition of a non strict $n$-groupoid. Then we associate to
any space $X$ an $n$-groupoid $\Pi _{_{n}}(X)$ which generalises the famous
Poincar\'e groupoid $\Pi _{_{1}}(X)$ and embodies the $n$-truncated homotopy
type of $X$. We also give a natural construction for the geometric realisation
of an $n$-groupoid and we conjecture that the functor geometric realisation is
an inverse up to equivalence to the functor $\Pi _{_{n}}(\ )$ from the category
of $n$-truncated topological spaces to the category $n$-Gr of $n$-groupoids.


title: Non-symplectic involutions of a K3 surface

abstract:
  We classify K3 surfaces with a non-symplectic finite automorphism of high
order. It is shown that such an automorphism cannot be of order 60, and for
each of the orders 38, 44, 48, 50, 54 and 66, there exists a unique K3 surface
with such an automorphism.


title: Applications of Koszul homology to numbers of generators and syzygies

abstract:
  Several spectral sequence techniques are used in order to derive information
about the structure of finite free resolutions of graded modules. These results
cover estimates of the minimal number of generators of defining ideals of
projective varieties. There are also investigations about the shifts and the
dimension of Betti numbers.


title: On Macaulayfication of certain quasi-projective schemes

abstract:
  The notion of Macaulayfication, which is analogous of the desingularization,
was introduced by Faltings in 1978 and he construct a Macaulayfication of
quasi-projective scheme whose non-Cohen-Macaulay locus is of dimension 0 or 1
by a characteristic free method. In this paper, we gave a Macaulayfication of a
quasi-projective scheme whose non-Cohen-Macaulay locus is of dimension 2. Of
course out method is independent of the characteristic.


title: Mathematics in and out of String Theory

abstract:
  We first give an exposition of how the Polyakov path integral for the bosonic
string produces a natural mapping class group invariant measure, $d(Poly)$, on
the Teichm\"uller space of Riemann surfaces of each fixed genus. The
description of $d(Poly)$ via the Mumford isomorphisms for determinant bundles
over Teichm\"uller space is also explained. We then report on our recent
results with I.Biswas and D.Sullivan, where we succeed in coherently fitting
together this Polyakov-Mumford construction over the universal direct limit,
$T_\infty$, of all the finite genus Teichm\"uller spaces. The virtual
automorphism group of the fundamental group of a compact surface (of arbitrary
genus $g > 1$) acts by automorphisms on $T_\infty$ as the ``commensurability
mapping class group''. Our entire construction is equivariant with respect to
this new modular group.


title: On Invariant Theory

abstract:
  Here we develop a technique of computing the invariants of $n-$ary forms and
systems of forms using the discriminants of corresponding multilinear forms
built of their partial derivatives, which should be cosidered as analogues of
classical discriminants and resultants for binary forms.


title: Remarks on numerical semigroups

abstract:
  We extend results on Weierstrass semigroups at ramified points of double
covering of curves to any numerical semigroup whose genus is large enough. As
an application we strengthen the properties concerning Weierstrass weights in
\cited{To}.


title: In\'egalit\'es de Morse et vari\'et\'es de Moishezon

abstract:
  The central topic of this thesis is the study of some properties of a class
of complex compact manifolds~: Moishezon manifolds. In the first part, we
generalize J.-P. Demailly's holomorphic Morse inequalities to the case of a
line bundle equipped with a metric with analytic singularities on an arbitrary
compact complex manifold. Our inequalities give an estimate of the cohomology
groups with values in the line bundle tensor powers twisted by the
corresponding sequence of multiplier ideal sheaves introduced by Nadel. As a
consequence, we obtain a necessary and sufficient analytic condition, invariant
by bimeromorphism, for a manifold to be Moishezon. In the second part, we use
Mori theory to analyze the structure of Moishezon manifolds with infinite
cyclic Picard group, with big canonical bundle, and which become projective
after one single blow-up with smooth center. We study the dimension and the
structure of the center of the blow-up. In dimension four, we show that this
locus is always a surface, and when the canonical bundle


title: Degenerations of elliptic curves and cusp singularities

abstract:
  We give more or less explicit equations for all two-dimensional cusp
singularities of embedding dimension at least 4. They are closely related to
Felix Klein's equations for universal curves with level n structure. The main
technical result is a description of the versal deformation of an n-gon in
$P^{n-1}$. The final section contains the equations for smoothings of simple
elliptic singularities (of multiplicity at most 9).


title: Lang maps and Harris's conjecture - a note in search for content

abstract:
  The Lang map, namely the universal dominant rational map to a variety of
general type, is constructed and briefly discussed in relation with arithmetic
conjectures of Harris, Lang and Manin. Existence of the Lang map follows from
the additivity of Kodaira dimension, but the fine structure depends on
conjectures on birational classification of algebraic varieties. Serious
applications of the Lang map are still being searched.


title: Deninger's conjecture on $L$-function of elliptic curves at $s=3$

abstract:
  I compute explicitly the regulator map on $K_4(X)$ for an arbitrary curve $X$
over a number field. Using this and Beilinson's theorem about regulators for
modular curves ([B2]) I prove a formula expressing the value of the
$L$-function $L(E,s)$ of a modular elliptic curve $E$ over $\Bbb Q$ at $s=3$ by
the double Eisenstein-Kronecker series.


title: Residue construction of Hecke algebras

abstract:
  Hecke algebras are usually defined algebraically, via generators and
relations. We give a new algebro-geometric construction of affine and
double-affine Hecke algebras (the former is known as the Iwahori-Hecke algebra,
and the latter was introduced by Cherednik [Ch1]). More generally, to any
generalized Cartan matrix A and a point q in a 1-dimensional complex algebraic
group \c we associate an associative algebra H. If A is of finite type and
\c=C^*, the algebra H is the affine Hecke algebra of the corresponding finite
root system. If A is of affine type and \c=C^* then H is, essentially, the
Cherednik algebra. The case \c=C corresponds to `degenerate' counterparts of
the above objects cosidered by Drinfeld [Dr] and Lusztig [L2]. Finally, taking
\c to be an elliptic curve one gets some new elliptic analogues of the affine
Hecke algebra.


title:
The Remark on Discriminants of K3 Surfaces Moduli as Sets of Zeros of
  Automorphic Forms

abstract:
  We show that for any $N>0$ there exists a natural even $n>N$ such that the
discriminant of moduli of K3 surfaces of the degree $n$ is not equal to the set
of zeros of any automorphic form on the corresponding IV type domain. We give
the necessary condition on a "condition $S\subset L_{K3}$ on Picard lattice of
K3'' for the corresponding moduli $\M_{S\subset L_{K3}}$ of K3 to have the
discriminant which is equal to the set of zeros of an automorphic form. We
conjecture that the set of $S\subset L_{K3}$ satisfying this necessary
condition is finite if $\rk S \le 17$. We consider this finiteness conjecture
as "mirror symmetric'' to the known finiteness results for arithmetic
reflection groups in hyperbolic spaces and as important for the theory of
Lorentzian Kac--Moody algebras and the related theory of automorphic forms.


title: Determinant of complexes and higher Hessians

abstract:
  Let $X \subset \Bbb P^r$ be a smooth algebraic curve in projective space,
over an algebraically closed field of characteristic zero. For each $m \in \Bbb
N$, the $m$-flexes of $X$ are defined as the points where the osculating
hypersurface of degree $m$ has higher contact than expected, and a hypersurface
$H \subset \Bbb P^r$ is called a $m$-Hessian if it cuts $X$ along its
$m$-flexes. When $X$ is a complete intersection, we give an expression for a
(rational) $m$-Hessian as the Div (in the sense of
Grothendieck-Knudsen-Mumford) of a complex of graded free modules naturally
associated to $X$. The construction of this complex involves relating sheaves
of differential operators on a scheme and a subscheme, and higher Euler
sequences on projective space.


title: Birational models for varieties of Poncelet curves

abstract:
  We consider Poncelet pairs $(S,C)$, where $S$ is a smooth conic and $C$ is a
degree$-c$ plane curve having the Poncelet property with respect to $S$. We
prove that for $c>4$ the projection $(S,C)\mapsto C$ is generically one-to-one
and use this to describe a birational model of the variety of Poncelet curves
for $c$ odd.


title: A sharp slope inequality for general stable fibrations of curves

abstract:
  Let M_g be the moduli space of stable curves of genus g >= 2. Let D_i be the
irreducible component of the boundary of M_g such that general points of D_i
correspond to stable curves with one node of type i. Let M_g^0 be the set of
stable curves that have at most one node of type i>0. Let d_i be the class of
D_i in Pic(M_g)_Q and h the Hodge class on M_g. In this paper, we will prove a
sharp slope inequality for general stable fibrations. Namely, if $C$ is a
complete curve on M_g^0, then ( (8g+4)h - g d_0 - \sum_{i=1}^{[g/2]} 4i(g-i)
d_i . C ) >= 0. As an application, we can prove effective Bogomolov's
conjecture for general stable fibrations.


title: Uniform boundedness for rational points

abstract:
  We extend an earlier result by Dan Abramovich, showing that a conjecture of
S. Lang's implies the existence of a uniform bound on the number of
$K$-rational points over all smooth curves of genus $g$ defined over $K$, where
$K$ is any number field of fixed degree $d$, and $g$ is an integer greater than
1. The bound depends only on the genus $g$ and the degree of the number field
$K$.


title: Higher genus symplectic invariants and sigma model coupled with gravity

abstract:
  We define higher genus Gromov-Witten invariants and establish a mathematical
theory of sigma model coupled with gravity over any semi-positive symplectic
manifolds. As applications, we verify the stablizing conjecture of symplectic
4-manifolds for simply connected elliptic surfaces and construct smooth
6-manifolds admitting infinitely many deformation classes of symplectic
structures.


title: Pieri's Formula Via Explicit Rational Equivalence

abstract:
  We present a new geometric proof of Pieri's formula, exhibiting an explicit
chain of rational equivalences from a suitable sum of distinct Schubert
varieties to the intersection of a Schubert variety with a special Schubert
variety. The geometry of these rational equivalences indicate a link to a
combinatorial proof of Pieri's formula using Schensted insertion. This version
includes an example to illustrate the proof.


title:
Sur les espaces de modules des fibres vectoriels de rang deux sur des
  hypersurfaces de P^3

abstract:
  Soit $X$ une hypersurface lisse de degr\'e $\delta \geq 4$ dans $\pp ^3$
telle que $Pic(X)=Z$. On d\'esigne par $M_X(c_2)$ l'espace de modules des
fibr\'es vectoriels sur $X$ de classes de Chern $c_1 = 0$ et $c_2$,
semi-stables par rapport au diviseur hyperplan. Nous contribuons ici \`a la
recherche de diff\'erentes composantes irr\'eductibles pour $c_2$ petit.
  On prouve que pour tout entier $c_2 \geq \delta ^3/4 - \delta ^2/2$ l'espace
des modules $M_X(c_2)$ contient une composante irr\'eductible r\'eduite de
dimension attendue. En utilisant un r\'esultat de O'Grady, on en d\'eduit que
pour tout entier $c_2$ tel que
  $$ {1/4}(\delta ^3-2\delta ^2) \leq c_2 \leq
  {1/3}(\delta ^3 - 9 \delta ^2 + 26 \delta - 3) $$ l'espace $M_X(c_2)$
poss\`ede au moins deux composantes irr\'eductibles de dimensions distinctes.
Pour $\delta \geq 27$, de tels entiers existent! D'autre part, on prouve que
pour $c_2 > {1/12}(13 \delta ^3 - 24 \delta ^2 + 8 \delta)$, le fibr\'e
g\'en\'eral de la bonne composante de $M_X(c_2)$ que nous construisons a
cohomologie naturelle. Plus g\'en\'eralement, on d\'emontre que pour toute
surface projective lisse $X$ telle que le fibr\'e canonique soit de la forme
$K_X = \Oo_X(k)$ o\`u $\Oo_X(1)$ est un fibr\'e ample, alors si $c_2$ est
suffisamment grand, le fibr\'e g\'en\'eral de l'unique composante de $M_X(c_2)$
a la cohomologie naturelle.


title: On a theorem of Harer

abstract:
  We use Harer's stability theorem to give another proof of his fundamental
theorem: The rank of the Picard group of the moduli space of smooth projective
curves of genus g > 2 equals one.


title: Holomorphic Rank Two Vector Bundles on Blow-ups

abstract:
  In this paper we study holomorphic rank two vector bundles on the blow up of
$ {\bf C}^2$ at the origin. A classical theorem of Birchoff and Grothendieck
says that any holomorphic vector bundle on the projective plane ${\bf P}^1$
splits into a sum of line bundles. If $E$ is a holomorphic vector bundle over
the blow up of $ {\bf C}^2$ at the origin, then the restriction of $E$ to the
exceptional divisor is a vector bundle over ${\bf P}^1$ and therefore splits.
Moreover we assume that $E$ is a rank two bundle that has zero first Chern
class. Hence its restriction to the exceptional divisor is of the form $ {\cal
O}(j) \oplus {\cal O}(-j) $ for some integer $j.$ We denote by ${\cal M}_j$ the
moduli space of equivalence classes (under holomorphic isomorphisms) of rank
two holomorphic vector bundles on the blow up of $ {\bf C}^2$ at the origin
whose restriction to the exceptional divisor is $ {\cal O}(j) \oplus {\cal
O}(-j) .$


title: The intrinsic normal cone

abstract:
  We suggest a construction of virtual fundamental classes of certain types of
moduli spaces.


title: Gromov-Witten invariants in algebraic geometry

abstract:
  Gromov-Witten invariants for arbitrary projective varieties and arbitrary
genus are constructed using the techniques from K. Behrend, B. Fantechi: The
intrinsic normal cone.


title: Super Curves, their Jacobians, and super KP equations

abstract:
  We study the geometry and cohomology of algebraic super curves, using a new
contour integral for holomorphic differentials. For a class of super curves
(``generic SKP curves'') we define a period matrix. We show that the odd part
of the period matrix controls the cohomology of the dual curve. The Jacobian of
a generic SKP curve is a smooth supermanifold; it is principally polarized,
hence projective, if the even part of the period matrix is symmetric. In
general symmetry is not guaranteed by the Riemann bilinear equations for our
contour integration, so it remains open whether Jacobians are always projective
or carry theta functions.
  These results on generic SKP curves are applied to the study of
algebro-geometric solutions of the super KP hierarchy. The tau function is
shown to be, essentially, a meromorphic section of a line bundle with trivial
Chern class on the Jacobian, rationally expressible in terms of super theta
functions when these exist. Also we relate the tau function and the Baker
function for this hierarchy, using a generalization of Cramer's rule to the
supercase.


title: Severi Degrees in Cogenus 4

abstract:
  In this paper, we give closed-form formulae for Severi degrees in cogenus 3
and 4 using Ran's method. These formulae coincide with those of I. Vainsencher
and for cogenus 3 case, that of J. Harris and R. Pandharipande. Another result
of this paper is that we calculate the degree of the polynomial
$N(\pi,\delta,d)$ in $d$, which is the degree of the locus of curves $C$ in P^2
with degree d having $\delta$ nodes and $C \cap L$ is of type $\pi$ to fixed
line L. Using this result, we also calculate the coefficients of two leading
terms of Severi polynomial, $N(\delta,d)$.


title: Global properties of families of plane curves

abstract:
  We describe degenerations of projective plane curves to curves containing a
fixed line $l$ as a component, and show that $H^1({\overline V}_{n,d,m}, {\Cal
O} (r))=0, r \in{\Bbb Z}$, where $V_{n,d,m}\subset {\Bbb P}^N (N = n(n+3)/2)$
is the subscheme consisting of irreducible plane curves having smooth contact
of order at least m with $l$ at a fixed point p of $l$ and d nodes and no other
singularities. Note: the notion of admissible schemes of plane curves,
introduced for the proof of the vanishing theorem, allows us to give a recipe
for calculating the Hilbert polynomial of ${\overline V}_{n,d}$ (see Sect. 4),
in particular the quantum cohomology of the plane.


title: Birational symplectic manifolds and their deformations

abstract:
  The known counterexamples to the global Torelli theorem for
higher-dimensional hyperkahler manifolds are provided by birational manifolds.
We address the question whether two birational hyperkahler manifolds (i.e.
irreducible symplectic) manifolds always define non-separated points in the
moduli space of marked manifolds. An affirmative answer is given for the cases
of Mukai's elementary transformations and birational correspondences which are
isomorphic in codimension two. The techniques are applied to show that the
moduli spaces of rank two sheaves on a K3 surface are deformation equivalent to
appropriate Hilbert schemes.


title: Space of conformal blocks in 4D WZW theory

abstract:
  We propose an algebro-geometric definition of the space of conformal blocks
in the four dimensional Wess-Zumino-Witten theory introduced by Losev.et al..


title: The canonical modifications by weighted blow-ups

abstract:
  In this paper we give a criterion for an isolated, hypersurface singularity
of dimension $n\ (\geq 2)$ to have the canonical modification by means of a
suitable weighted blow-up. Then we give a counter example to the following
conjecture by Reid-Watanabe: For a 3-dimensional, isolated, non-canonical,
log-canonical singularity $(X,x)$ of embedded dimension 4, there exists an
embedding $(X,x)\subset ({\bC}^4, 0)$ and a weight ${\bw}=(w_0,w_1,\ldots
,w_n)$, such that the ${\bw}$-blow-up gives the canonical modification of
$(X,x)$.


title: Bounds for Seshadri Constants

abstract:
  In the paper we present an alternative approach to the boundedness of
Seshadri constants (which measure the local positivity) of nef and big line
bundles at a general point of a complex--projective variety. Our approach is
based on the construction of divisors in certain adjoint linear systems having
isolated singularities of high order. A variant of the method gives bounds for
Seshadri constants valid at arbitrary points of a smooth variety; these bounds,
however, depend on the line bundle and the positivity of the tangent bundle of
the variety in question.


title: Principle bundles admitting a holomorphic structure

abstract:
  Let $M$ be a compact connected K\"ahler manifold and let ${\E}_{l-1}$ be the
smallest term in the Harder-Narasimhan filtration of its tangent bundle. Let
$G$ be an affine algebraic reductive group over $\C$. We prove the following
result: If $M$ satisfies the condition that $\deg (T/{\E}_{l-1}) \geq 0$, then
a holomorphic principal $G$-bundle $P$ on $M$ admitting a compatible
holomorphic connection is semistable. Moreover, if $\deg (T/{\E}_{l-1}) >0$,
then such a bundle $P$ actually admits a compatible flat $G$-connection.


title: Geometry of families of nodal curves on rational surfaces

abstract:
  Let P^2_r be the projective plane blown up at r generic points. Denote by
E_0,E_1,...,E_r the strict transform of a generic straight line on P^2 and the
exceptional divisors of the blown-up points on P^2_r respectively. We consider
the variety V of all irreducible curves C in |dE_0-d_1E_1-...-d_rE_r| with k
nodes as the only singularities and give asymptotically nearly optimal
sufficient conditions for its smoothness, irreducibility and non-emptyness.
Moreover, we extend our conditions for the smoothness and the irreducibility on
families of reducible curves. For r<10 we give the complete answer concerning
the existence of nodal curves in V.


title: Volumes of hyperbolic manifolds and mixed Tate motives

abstract:
  Two different constructions of an invariant of an odd dimensional hyperbolic
manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The
volume of the manifold is equal to the value of the Borel regulator on that
element. The scissor congruence groups in non euclidian geometries are studied
and their relationship with algebraic K-theory of the field of complex numbers
is discussed.


title: The Space of Triangles, Vanishing Theorems, and Combinatorics

abstract:
  We consider compactifications of the space of triples of distinct points in
projective $n$-space. One such space is a singular variety of configurations of
points and lines; another is the smooth compactification of Fulton and
MacPherson; and a third is the triangle space of Schubert and Semple. We
compute the sections of line bundles on these spaces, and show that they are
equal as GL(n) representations to the generalized Schur modules associated to
``bad'' generalized Young diagrams with three rows (Borel-Weil theorem). On the
one hand, this yields Weyl-type character and dimension formulas for the Schur
modules; on the other, a combinatorial picture of the space of sections.
Cohomology vanishing theorems play a key role in our analysis.


title: Compactifications of families of curves

abstract:
  We construct a good compactification of the variety of irreducible projective
plane curves of degree n with d nodes and no other singularities.


title: Toward Clemens' Conjecture in degrees between 10 and 24

abstract:
  We introduce and study a likely condition that implies the following form of
Clemens' conjecture in degrees $d$ between 10 and 24: given a general quintic
threefold $F$ in complex $\IP^4$, the Hilbert scheme of rational, smooth and
irreducible curves $C$ of degree $d$ on $F$ is finite, nonempty, and reduced;
moreover, each $C$ is embedded in $F$ with balanced normal sheaf
$\O(-1)\oplus\O(-1)$, and in $\IP^4$ with maximal rank.


title: Geometry of Deligne cohomology

abstract:
  It is well known that degree two Deligne cohomology groups can be identified
with groups of isomorphism classes of holomorphic line bundles with
connections. There is also a geometric description of degree three Deligne
cohomology, due to J-L. Brylinski and P. Deligne, in terms of gerbes with
connective structures and curvings. This paper gives a geometric interpretation
of Deligne cohomology of all degrees, in terms of equivalence classes of higher
line bundles with $k$-connections. It is also shown that the classical
Abel-Jacobi isomorphism $Pic^0(X) \cong J(X)$ generalizes to the isomorphism
between groups of equivalence classes of topologically trivial 1-holomorphic
higher line bundles with $k$-connections and Griffiths intermediate Jacobians.


title: Singularities of Pairs

abstract:
  These are the revised notes of my lectures at the 1995 Santa Cruz summer
institute. Changes from the Jan.26,1996 version: Major: 7.1--2; Medium: 3.7,
3.11, 5.3.3, 7.9.2, 9.8.


title: On Cohomology of the Square of an Ideal Sheaf

abstract:
  For a smooth subvariety $X\subset\Bbb P^N$, consider (analogously to
projective normality) the vanishing condition $H^1(\Bbb P^N,\Cal I^2_X(k))=0$,
$k\ge3$. This condition is shown to be satisfied for all sufficiently large
embeddings of a given $X$, and for a Veronese embedding of $\Bbb P^n$. For
$C\subset\Bbb P^{g-1}$, the canonical embedding of a non-hyperelliptic curve,
this condition guarantees the vanishing of some obstruction groups to
deformations of the cone. Recall that the tangents to deformations are dual to
the cokernel of the Gaussian-Wahl map.
  \proclaim{Theorem} Suppose the Gaussian-Wahl map of $C$ is not surjective and
the vanishing condition is fulfilled. Then $C$ is {\bf extendable}: it is a
hyperplane section of a surface in $\Bbb P^g$ not the cone over
$C$.\endproclaim
  Such a surface is a K3 if smooth, but it could have serious singularities.
  \proclaim{Theorem} For a general curve of genus $\ge3$, this vanishing holds.
\endproclaim
  \proclaim{Conjecture} If the Clifford index is $\ge3$, this vanishing holds.
\endproclaim


title: Algebraic Chern Simons Theory

abstract:
  An analog of Chern-Simons theory is developed in an algebro-geometric
setting.


title: Flatness of families induced by hypersurfaces on flag varieties

abstract:
  We answer a question posed by S. Kleiman concerning flatness of the family of
complete quadrics. We also show that any flat family of hypersurfaces on
Grassmann varieties induces a flat family of intersections with the
corresponding flag variety.


title:
A Hitchin-Kobayashi Correspondence for Coherent Systems on Riemann
  Surfaces

abstract:
  A `coherent system' $(\Cal E,V)$, consists of a holomorphic bundle plus a
linear subspace of its space of holomorphic sections. Based on the usual notion
in Geometric Invariant Theory, a notion of slope stability has been defined for
such objects (by Le Potier, and also by Rhagavendra and Vishwanath). In this
paper we show that stability in this sense is equivalent to the existence of
solutions to a certain set of gauge theoretic equations. One of the equations
is essentially the vortex equation (i.e. the Hermitian-Einstein equation with
an additional zeroth order term), and the other is an orthonormality condition
on a frame for the subspace $V\subset H^0(\Cal E)$.


title: Alexander Stratifications of Character Varieties

abstract:
  There is a natural stratification of the character variety of a finitely
presented group coming from the jumping loci of the first cohomology of
one-dimensional representations. Equations defining the jumping loci can be
effectively computed using Fox calculus. In this paper, we give an exposition
of Fox calculus in the language of group cohomology and in the language of
finite abelian coverings of CW complexes. Work of Simpson, Arapura and others
show that if $\Gamma$ is the fundamental group of a compact K\"ahler manifold,
then the strata are finite unions of translated affine subtori. If follows that
for K\"ahler groups the jumping loci must be defined by binomial ideals. We
discuss properties of the jumping loci of general finitely presented groups and
apply the ``binomial criterion" to obtain new obstructions for one-relator
groups to be K\"ahler.


title: Real enumerative geometry and effective algebraic equivalence

abstract:
  We describe an approach to the question of finding real solutions to problems
of enumerative geometry, in particular the question of whether a problem of
enumerative geometry can have all of its solutions be real. We give some
methods to infer one problem can have all of its solutions be real, given that
a related problem does. These are used to show many Schubert-type enumerative
problems on some flag manifolds can have all of their solutions be real.


title: Chapters on algebraic surfaces

abstract:
  This is a first graduate course in algebraic geometry. It aims to give the
student a lift up into the subject at the research level, with lots of
interesting topics taken from the classification of surfaces, and a
human-oriented discussion of some of the technical foundations, but with no
pretence at an exhaustive treatment. The early chapters introduce topics that
are useful throughout projective and algebraic geometry, make little demands,
and lead to fun calculations. The intermediate chapters introduce elements of
the technical language gradually, whereas the later chapters get into the
substance of the classification of surfaces. Special features include the
theory of minimal models of surfaces via Mori theory, a complete selfcontained
proof of the theorems on classification of surfaces, and a clean treatment of
the foundational results on rational and elliptic Gorenstein surface
singularities.
  Contents:
  Chapter 1. The cubic surface p.4 Exercises to Chapter 1 p.12
  Chapter 2. Rational scrolls p.14 Exercises to Chapter 2 p.23
  Chapter A. Curves on surfaces and intersection numbers p.26 Exs to Ch A p.35
  Chapter B. Sheaves and coherent cohomology p.37 Exercises to Chapter B p.47
  Chapter C. Guide to the classification of surfaces 51
  Chapter 3. K3s p.63 Exercises to Chapter 3 p.76
  Chapter 4. Surfaces and singularities p.80 Exercises to Chapter 4 p.106
  Chapter D. Minimal models of surfaces via Mori theory p.110
  Chapter E. Proof of the classification of surfaces p.121
  References p.146


title:
Virtual moduli cycles and Gromov-Witten invariants of algebraic
  varieties

abstract:
  We introduce a method of constructing the virtual cycle of any scheme
associated with a tangent-obstruction complex. We apply this method to
constructing the virtual moduli cycle of the moduli of stable maps from
n-pointed genus g curves to any smooth projective variety. As an application,
we give an algebraic definition of GW-invariants for any smooth projective
variety.


title: Spin Verlinde spaces and Prym theta functions

abstract:
  Theta functions of level n on the principally polarised Prym varieties of an
algebraic curve are dual to sections of the orthogonal theta line bundle on the
moduli space of Spin(n)-bundles over the curve. As a by-product of our
computations we also note that when n is odd the pfaffian line bundle on moduli
space has a basis of sections labelled by the even theta characteristics of the
curve.


title: On the monodromy at infinity of a polynomial map, II

abstract:
  In the last years a lot of work has been concentrated on the study of the
behaviour at infinity of polynomial maps. This behaviour can be very
complicated, therefore the main idea was to find special classes of polynomial
maps which have, in some sense, nice properties at infinity. In this paper, we
completely determine the complex algebraic monodromy at infinity for a special
class of polynomial maps (which is complicated enough to show the nature of the
general problem).


title: Non-abelian monopoles and vortices

abstract:
  The Seiberg-Witten equations are defined on certain complex line bundles over
smooth oriented four manifolds. When the base manifold is a complex Kahler
surface, the Seiberg-Witten equations are essentially the Abelian vortex
equations. Using known non-abelian generalizations of the vortex equations as a
guide, we explore some non-abelian versions of the Seiberg-Witten equations. We
also make some comments about the differences between the vortex equations that
have previously appeared in the literature and those that emerge as Kahler
versions of Seiberg-witten type equations.


title: Residues and Differential Operators on Schemes

abstract:
  Beilinson Completion Algebras (BCAs) are generalizations of complete local
rings, and have a rich algebraic-analytic structure. These algebras were
introduced in my paper "Traces and Differential Operators over Beilinson
Completion Algebras", Compositio Math. 99 (1995). In the present paper BCAs are
used to give an explicit construction of the Grothendieck residue complex on an
algebraic scheme. This construction reveals new properties of the residue
complex, and in particular its interaction with differential operators.
Applications include: (i) results on the algebraic structure of rings of
differential operators; (ii) an analysis of the niveau spectral sequence of De
Rham homology; (iii) a proof of the contravariance of De Rham homology w.r.t.
etale morphisms; (iv) an algebraic description of the intersection cohomology
D-module of a curve.


title: Nodal curves on surfaces of general type

abstract:
  We study the family of irreducible curves with $\delta$ nodes belonging to a
free linear system $|C|$ with smooth general member on a surface $S$ such that
$|K_S|$ is ample. Under the assumption that $C$ is numerically equivalent to
$pK_S$, $p\ge 3$ rational, we give an upper bound on $\delta$ ensuring that the
family is smooth of codimension $\delta$ in $|C|$ (plus another slightly
different result of the same type). We give examples showing that the bound is
sharp. Some related results concerning linear normality of nodal curves are
proved.


title: Rational points on some Fano cubic bundles

abstract:
  We consider some families of smooth Fano hypersurfaces $X_{n+2}$ in ${\bf
P}^{n+2} \times {\bf P}^3$ given by a homogeneous polynomial of bidegree
$(1,3)$. For these varieties we obtain lower bounds for the number of
$F$-rational points of bounded anticanonical height in arbitrary nonempty
Zariski open subset $U \subset X_{n+2}$. These bounds contradict previous
expectations about the distribution of $F$-rational points of bounded height on
Fano varieties.


title: Reduction of abelian varieties

abstract:
  We study semistable reduction and torsion points of abelian varieties. In
particular, we give necessary and sufficient conditions for an abelian variety
to have semistable reduction. We also study N\'eron models of abelian varieties
with potentially good reduction and torsion points of small order. We study
some invariants that measure the extent to which an abelian variety with
potentially good reduction fails to have good reduction.


title:
Semicontinuity for representations of one-dimensional Cohen-Macaulay
  rings

abstract:
  We show that the number of parameters for CM-modules of prescribed rank is
semi-continuous in families of CM rings of Krull dimension 1. This transfers a
result of Knoerrer from the commutative to the not necessarily commutative
case. For this purpose we introduce the notion of ``dense subrings'' which
seems rather technical but, nevertheless, useful. It enables the construction
of ``almost versal'' families of modules for a given algebra and the definition
of the ``number of parameters''. The semi--continuity implies, in particular,
that the set of so-called ``wild algebras'' in any family is a countable union
of closed subsets. A very exciting problem is whether it is actually closed,
hence whether the set of tame algebras is open. However, together with the
results of a former paper of the authors the semi-continuity implies that tame
is indeed an open property for curve singularities (commutative CM rings). An
analogous procedure leads to the semicontinuity of the number of parameters in
other cases, like representations of finite dimensional algebras or finite
dimensional bimodules.


title: Conics Touching a Quartic Surface with 13 Nodes

abstract:
  Motivated by questions occuring in the construction of certain twistor spaces
the parameter space of conics tangent to a given quartic is investigated. For a
given real quartic surface in complex $\PP ^3$ that has exactly 13 ordinary
nodes as singularities this parameter space is studied. In particular, its
irreducible components are described.


title: Topology of Conic Bundles - II

abstract:
  For conic bundles on a smooth variety (over a field of characteristic $\ne
2$) which degenerate into pairs of distinct lines over geometric points of a
smooth divisor, we prove a theorem which relates the Brauer class of the
non-degenerate conic on the complement of the divisor to the covering class
(Kummer class) of the 2-sheeted cover of the divisor defined by the degenerate
conic, via the Gysin homomorphism in etale cohomology. This theorem is the
algebro-geometric analogue of a topological result proved earlier.


title: P-Resolutions of Cyclic Quotients from the Toric Viewpoint

abstract:
  P-resolutions of two-dimensional, cyclic quotient singularities have been
introduced to study deformation theory. Those P-resolutions (as well as the
singularities themselves) are toric varieties. In the present paper we give a
straight, elementary description of them just by their defining fans.


title:
Formulas for Lagrangian and orthogonal degeneracy loci; the
  Q-polynomials approach

abstract:
  Let V be a vector bundle on a scheme X endowed with a nondegenerate
symplectic or orthogonal form. Let G be a Grassmannian bundle parametrizing
maximal isotropic subbundles of V. The main goal of the paper is to give
formulas for the classes of the loci in G parametrizing those subbundles which
intersect successive members of of a fixed flag of isotropic subbundles of V in
dimensions greater than 1,2,3,... . The formulas which we give are quadratic
expressions in Q-polynomials of the tautological bundle on G and flag
S-polynomials of the members of the flag. These formulas globalize to
Lagrangian and orthogonal degeneracy loci. They have especially nice form for
the loci of points in X where two maximal isotropic subbundles intersect in
dimension exceeding a given number. (A problem for finding formulas for these
loci was posed by J. Harris several years ago.) In this case, our formulas are
simple quadratic expressions of Q-polynomials applied to E and F. The work
generalizes formulas given in [P.Pragacz, Algebro- geometric applications of
Schur S- and Q-polynomials, SLN 1478 (1991), 130-191].
  One of its applications is computation of the classes of Brill-Noether loci
in Pryms in [De Concini, P. Pragacz, On the class of Brill-Noether loci for
Prym varieties, Math. Ann. 302 (1995), 687-697].


title: The Eisenbud-Koh-Stillman Conjecture on Linear Syzygies

abstract:
  It is proved, as was conjectured by Eisenbud-Koh-Stillman, that for a
finitely generated graded module $M$ over the symmetric algebra $S(V)$, if the
Koszul group ${\cal K}_{p,0}(M,V)\ne 0$, then the set of rank 1 relations in
$M_0\otimes V$ has dimension $\ge p$. The method is by using ``exterior minors"
to study syzygies of an ideal derived from the Koszul class in the exterior
algebra. As a consequence, a conjecture of Lazarsfeld and myself that a set $Z$
of $2r+1-p$ points in ${\bf P}^r$ for which property $N_p$ fails has a subset
$Z'$ of at least $2{\rm dim}(L) +2-p$ points lying on a linear space $L$ such
that property $N_p$ fails for $Z'$.


title: Bounds on cohomology and Castelnuovo-Mumford regularity

abstract:
  The Castelnuovo-Mumford regularity reg(X) of a projective scheme X was
introduced by Mumford by generalizing ideas of Castelnuovo. The interest in
this concept stems partly from the fact that X is m-regular if and only if for
every p \geq 0 the minimal generators of the p-th syzygy module of the defining
ideal I of X occur in degree \leq m + p. There are some bounds in the case that
X is a locally Cohen-Macaulay scheme. The aim of this paper is to extend and
improve these results for so-called (k,r)-Buchsbaum schemes. In order to prove
our theorems, we need to apply a spectral sequence. We conclude by describing
two sharp examples and open problems.


title: Towards a theory of arithmetic degrees

abstract:
  The aim of this paper is to start a systematic investigation of the
arithmetic degree of projective schemes as introduced by D. Bayer and D.
Mumford. One main theme concerns itself with the behaviour of this arithmetic
degree under hypersurface sections. The notion of arithmetic degree involves
the new concept of length-multiplicity of embedded primary ideals. Therefore it
is much harder to control the arithmetic degree under a hypersurface section
than in the case for the classical degree theory. Nevertheless it has important
and interesting applications. We describe such applications to the
Castelnuovo-Mumford regularity and to Bezout-type theorems.


title: Sheaves with connection on abelian varieties

abstract:
  The Fourier-Mukai transform is lifted to the derived category of sheaves with
connection on abelian varieties. The case of flat connections (D-modules) is
discussed in detail.


title: Even sets of nodes are bundle symmetric

abstract:
  Let k be an algebraically closed field of characteristic p different from 2,
and let F be a nodal surface of degree d in the projective 3-space P over k
(i.e. the singularities of F are only ordinary quadratic, nodes for short). Let
N be a subset of the set of nodes of F: then N is said to be n/2-even for n=0,1
if the following condition (*) holds. Namely, let F' be a minimal resolution of
the singularities of F, let N' be the inverse image of N in F', let H be the
inverse image of a plane section of F. Then N is n/2 if: (*) the class of N'+nH
is 2-divisible in Pic(F'). In this paper we prove the following
characterization of even sets, which has been conjectured by W. Barth and the
second author in 1979. Let F be as above. Then the n/2-even set of nodes N on F
are exactly the bundle symmetric sets, i.e. the degeneracy loci of symmetric
maps s:E*(-d-n)->E of a suitable locally free sheaf E over P (i.e. F is the
locus where rk(s)<rkE, N is the locus where rk(s)=rkE-2). We also give a
classification of even sets in degree d=4,5.


title:
Images of $\ell$-adic representations and automorphisms of abelian
  varieties

abstract:
  Suppose $F$ is either a global field or a finitely generated extension of
${\mathbf Q}$, $A$ is an abelian variety over $F$, and $\ell$ is a prime not
equal to the characteristic of $F$. Let $Z$ denote the center of the
endomorphism algebra of $A$. Let $G$ denote the group of ${\mathbf
Q}_\ell$-points of the identity connected component of the Zariski closure of
the image of the $\ell$-adic representation associated to $A$. We prove the
$\ell$-independence of the intersection of $G$ with the torsion subgroup of
$Z$. Our results provide evidence in the direction of the Mumford-Tate
Conjecture.


title: Connectedness extensions for abelian varieties

abstract:
  Suppose $A$ is an abelian variety over a field $F$, and $\ell$ is a prime not
equal to the characteristic of $F$. Let $F_{\Phi,\ell}(A)$ denote the smallest
extension of $F$ such that the Zariski closure of the image of the $\ell$-adic
representation associated to $A$ is connected. Serre introduced this field, and
proved that when $F$ is a finitely generated extension of ${\mathbf Q}$,
$F_{\Phi,\ell}(A)$ does not depend on the choice of $\ell$. In this paper we
study extensions $F_{\Phi,\ell}(B)/F$ for twists $B$ of a given abelian
variety, especially when the abelian varieties are of Weil type.


title:
Seiberg-Witten invariants for manifolds with $b_+=1$, and the universal
  wall crossing formula

abstract:
  In this paper we describe the Seiberg-Witten invariants, which have been
introduced by Witten, for manifolds with $b_+=1$. In this case the invariants
depend on a chamber structure, and there exists a universal wall crossing
formula. For every K\"ahler surface with $p_g=0$ and $q$=0, these invariants
are non-trivial for all $Spin^c(4)$-structures of non-negative index.


title: Transformation de Fourier generalisee

abstract:
  In this paper I construct a geometric transformation for generalized
1-motives which extends the Fourier-Mukai transformation for O-Modules on
abelian varieties, the geometric Fourier transformation for D-Modules on vector
spaces and the geometric Mellin transformation for D-Modules on tori. In
particular, I construct an equivalence of triangulated categories between the
derived category of quasi-coherent D-Modules on an abelian variety and the
derived category of quasi-coherent O-Modules on the universal extension of the
dual abelian variety. This equivalence has also been obtained by Mitchell
Rothstein.


title: Weak approximation and Manin group of R-equivalences

abstract:
  We extend an exact sequence of Colliot-Thelene and Sansuc which connects
arithmetic and birational invariants of tori over a number field to one for
arbitrary connected linear algebraic groups.


title: p-adic abelian integrals and commutative Lie groups

abstract:
  The aim of this paper is to propose an ``elementary" approach to Coleman's
theory of p-adic abelian integrals. Our main tool is a theory of commutative
p-adic Lie groups (the logarithm map); we use neither dagger analysis nor
Monsky-Washnitzer cohomology theory. Notice that we also treat the case of bad
reduction. A preliminary version of this paper appeared as Expos\'e 9 dans
``Problemes Diophantiens 88-89" (D. Bertrand, M. Waldschmidt), Publ. Math.
Univ. Paris VI 90(1990), 15 pp.


title:
Weight systems for toric Calabi-Yau varieties and reflexivity of Newton
  polyhedra

abstract:
  According to a recently proposed scheme for the classification of reflexive
polyhedra, weight systems of a certain type play a prominent role. These weight
systems are classified for the cases $n=3$ and $n=4$, corresponding to toric
varieties with K3 and Calabi--Yau hypersurfaces, respectively. For $n=3$ we
find the well known 95 weight systems corresponding to weighted $\IP^3$'s that
allow transverse polynomials, whereas for $n=4$ there are 184026 weight
systems, including the 7555 weight systems for weighted $\IP^4$'s. It is proven
(without computer) that the Newton polyhedra corresponding to all of these
weight systems are reflexive.


title: Equivariant intersection theory

abstract:
  In this paper we develop an equivariant intersection theory for actions of
algebraic groups on algebraic schemes. The theory is based on our construction
of equivariant Chow groups. They are algebraic analogues of equivariant
cohomology groups which satsify the formal properties of ordinary Chow groups.
In addition, they enjoy many of the properties of equivariant cohomology. The
principal results are: (1) We prove the existence of canonical intersection
products on the Chow groups of geometric quotients of smooth varieties- even
when the stabilizers of geometric points are non-reduced. (2) We construct a
Todd class map from equivariant $K$-theory of coherent sheaves to a completion
of equivariant Chow groups, and prove that a completion of equivariant
$K$-theory is isomorphic to the completion of equivariant Chow groups. (3) We
prove a localization theorem for torus actions and use it to give a
characteristic free proof of the Bott residue formula for actions of tori on
complete smooth varieties.


title: A Note on the Cohomology of Moduli of Rank Two Stable Bundles

abstract:
  The rational cohomology of the moduli space of rank two, odd degree stable
bundles over a curve (of genus g > 1) has been studied intensely in recent
years and in particular the invariant subring generated by Newstead's
generators alpha, beta, gamma. Several authors have independently found a
minimal complete set of relations for this subring. Their methods are very
different from the methods originally employed by Kirwan to prove Mumford's
conjecture -- that relations derived from the vanishing Chern classes of a
particular rank 2g-1 bundle are a complete set of relations for the entire
cohomology ring. This note contains two theorems which readily follow from
Kirwan's original calculations. We rederive the above result showing that the
first three invariant Mumford relations generate the relation ideal of the
invariant subring. Secondly we prove a stronger version of Mumford's conjecture
and show that the relations coming from the first vanishing Chern class
generate the relation ideal of the entire cohomology ring as a
Q[alpha,beta]-module. (Only a few typos have been amended in this revised
version).


title:
The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody
  algebras

abstract:
  We find automorphic corrections for the Lorentzian Kac--Moody algebras with
the simplest generalized Cartan matrices of rank 3:
  A_{1,0} =
  2 0 -1
  0 2 -2
  -1 -2 2
  and
  A_{1,I} =
  2 -2 -1
  -2 2 -1
  -1 -1 2
  For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form
\chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30
with respect to a 2-congruence subgroup. We find infinite product or sum
expansions for these forms. Our method of construction of \chi_{35} leads to
the direct construction of Siegel modular forms by infinite product expansions,
whose divisors are the Humbert surfaces with fixed discriminants. Existence of
these forms was proved by van der Geer in 1982 using some geometrical
consideration. We announce a list of all hyperbolic symmetric generalized
Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a
lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.


title: The Hilbert Schemes of Degree Three Curves are Connected

abstract:
  In this paper we show that the Hilbert scheme $H(3,g)$ of locally
Cohen-Macaulay curves in $\Pthree$ of degree three and genus $g$ is connected.
In contrast to $H(2,g)$, which is irreducible, $H(3,g)$ generally has many
irreducible components (roughly $-g/3$ of them). To show connectedness, we
classify the curves (giving particular attention to the triple lines),
determine the irreducible components, and give flat families over $\Aone$ to
show that the components meet. As a byproduct, we find that there are curves
which lie in the closure of each irreducible component.


title: Vanishing theorems for ample vector bundles

abstract:
  We prove a general vanishing theorem for the cohomology of products of
symmetric and skew-symmetric powers of an ample vector bundle on a smooth
complex projective variety. Special cases include an extension of classical
theorems of Griffiths and Le Potier to the whole Dolbeault cohomology, and an
answer to a problem raised by Demailly. An application to degeneracy loci is
given.


title: On curves over finite fields with many rational points

abstract:
  We study arithmetical and geometrical properties of {\it maximal curves},
that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of
$\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a
hypothesis on non-gaps at rational points we prove that maximal curves are
$\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.


title: Maximal Degeneracy Points of GKZ Systems

abstract:
  Motivated by mirror symmetry, we study certain integral representations of
solutions to the Gel'fand-Kapranov-Zelevinsky(GKZ) hypergeometric system. Some
of these solutions arise as period integrals for Calabi-Yau manifolds in mirror
symmetry. We prove that for a suitable compactification of the parameter space,
there exists certain special boundary points, which we called maximal
degeneracy points, at which all but one solutions become singular.


title: An intersection number for the punctual Hilbert scheme of a surface

abstract:
  Let S be a smooth projective surface, and consider the following two
subvarieties of the Hilbert scheme parameterizing closed subschemes of S of
length n:
  A = {subschemes with support in a fixed point of S}
  B = {subschemes with support in one (variable) point of S}
  A and B have complementary dimensions in the Hilbert scheme. We prove that
the intersection number [A].[B] = n(-1)^(n-1), answering a question by H.
Nakajima.


title: Espaces abstraits de morphismes et mutations

abstract:
  A transformation of morphisms of sheaves, called mutation, is used to build
new moduli spaces of morphisms.


title:
Singularities of Theta Divisors, and the Birational Geometry of
  Irregular Varieties

abstract:
  The purpose of this paper is to show how the generic vanishing theorems of M.
Green and the second author can be used to settle several questions and
conjectures concerning the geometry of irregular complex projective varieties.
First, we prove a conjecture of Arbarello and DeConcini characterizing
principally polarized abelian varieties whose theta divisors are singular in
codimension one. Secondly, we study the holomorphic Euler characteristics of
varieties of general type having maximal Albanese dimension: we verify a
conjecture of Kollar for subvarieties of abelian varieties, but show that it
fails in general. Finally, we give a surprisingly simple new proof of a
fundamental theorem of Kawamata concerning the Albanese mapping of projective
varieties of Kodaira dimension zero.


title: Smoothness, Semistability, and Toroidal Geometry

abstract:
  We provide a new proof of the following result:
  Let $X$ be a variety of finite type over an algebraically closed field $k$ of
characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a
modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular
variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings.
  Needless to say, this theorem is a weak version of Hironaka's well known
theorem on resolution of singularities. Our proof has the feature that it
builds on two standard techniques of algebraic geometry: semistable reduction
for curves, and toric geometry.
  Another proof of the same result was discovered independently by F. Bogomolov
and T. Pantev. The two proofs are similar in spirit but quite different in
detail.


title: Weak Hironaka theorem

abstract:
  The purpose of this note is to give a simple proof of the following theorem:
Let $X$ be a normal projective variety over an algebraically closed field $k$,
$\op{char} k = 0$ and let $D \subset X$ be a proper closed subvariety of $X$.
Then there exist a smooth projective variety $M$, a strict normal crossings
divisor $R \subset M$ and a birational morphism $f : M \to X$ with $f^{-1} D =
R$. The method of proof is inspired by A.J. de Jong alteration ideas. We also
use a multidimensional version of G.Belyi argument which allows us to simplify
the shape of a ramification divisor. By induction on the dimension of $X$ the
problem is reduced to resolving toroidal singularities. This process however is
too crude and does not permit any control over the structure of the birational
map $f$. A different proof of the same theorem was found independently by D.
Abramovich and A.J. de Jong. The approach is similar in both proofs but they
seem to be rather different in details.


title: Calabi-Yau Varieties and Pencils of K3 Surfaces

abstract:
  In this note, we give a list of Calabi-Yau hypersurfaces in weighted
projective 4-spaces with the property that a hypersurface contains naturally a
pencil of K3 variety. For completeness we also obtain a similar list in the
case K3 hypersurfaces in weighted projective 3-spaces. The first list
significantly enlarges the list of K3-fibrations of \KlemmLercheMayr~ which has
been obtained on some assumptions on the weights. Our lists are expected to
correspond to examples of the so-called heterotic-type II duality
\KachruVafa\KachruSilverstein.


title: Equivariant Gromov - Witten Invariants

abstract:
  We develop general theory of equivariant quantum cohomology for ample Kahler
manifolds and prove the mirror conjecture for projective complete
intersections.


title: On freeness theorem of the adjoint bundle on a normal surface

abstract:
  We extend Reider's freeness criterion to normal surfaces of characteristic 0.
  Let Y be a normal surface.
  Let D be a nef divisor on Y such that K_Y+D is a Cartier divisor.
  Let x be a point on Y.
  If x is a base point of |K_Y+D| and D^2>\delta_x (\delta_x is determined by
x, \delta_x <= 4) then there exists a non zero effective divisor E on Y passing
through x such that
  0 <= DE <= \delta_x /2,
  DE - \delta_x /4 <= E^2 <= (DE)^2 / D^2,
  E^2 < 0 if DE=0.


title:
Higher dimensional examples of manifolds whose adjoint bundles are not
  spanned

abstract:
  Let $(X,L)$ be an $n$-dimensional polarized variety. Fujita's conjecture says
that if $L^n>1$ then the adjoint bundle $K_X+nL$ is spanned and $K_X+(n+1)L$ is
very ample. There are some examples such that $K_X+nL$ is not spanned or
$K_X+(n+1)L$ is not very ample. These are $(\P^n,\O(1))$, hypersurface $M$ of
degree $6$ in weighted projective space $\P(3,2,1,1,\cdots ,1)$ with $\O_M(1)$
and numerically Godeaux surface etc. Numerically Godeaux surface is the
quotient space of a Fermat type hypersurface of degree $5$ in $\P^3$ by an
action of order $5$. These examples are not so much. We construct new examples
for any dimention.


title: d-gonality of modular curves and bounding torsions

abstract:
  We study the problem of $d$-gonality of the modular curve $X_0(N)$. As a
result, we can give an upperbound of the level $N$ by means of $d$. This
generalizes Ogg's result on hyperelliptic modular curves ($d = 2$). As a
corollary of this result, we prove an analogue of the strong Uniform
Boundedness Conjecture for elliptic curves defined over the function fields of
curves. If a base curve is $d$-gonal, we can bound orders of torsions of
Mordell-Weil groups in terms of $d$ uniformly.


title: Seiberg-Witten invariants for manifolds with $b_+=1$

abstract:
  In this paper we describe the Seiberg-Witten invariants, which have been
introduced by Witten, for manifolds with $b_+=1$. In this case the invariants
depend on a chamber structure, and there exists a universal wall crossing
formula. We take into account the contribution of the 1-homology of the
base-manifold. For every K\"ahler surface with $p_g=0$ and $q$=0, these
invariants are non-trivial for all $Spin^c(4)$-structures of non-negative
index.


title:
Higher Weil-Petersson Volumes of Moduli Spaces of Stable $n$-pointed
  Curves

abstract:
  Moduli spaces of compact stable $n$-pointed curves carry a hierarchy of
cohomology classes of top dimension which generalize the Weil-Petersson volume
forms and constitute a version of Mumford classes. We give various new formulas
for the integrals of these forms and their generating functions. We also
discuss their relation to the Kuenneth formula in quantum cohomology.


title: Threefolds in $\Bbb P^5$ with a 3-dimensional family of plane curves

abstract:
  A classification theorem is given of smooth threefolds of $\Bbb P^5$ covered
by a family of dimension at least three of plane integral curves of degree
$d\geq 2.$ It is shown that for such a threefold $X$ there are two
possibilities: \item{(1)} $X$ is any threefold contained in a hyperquadric;
\item{(2)} $d\leq 3$ and $X$ is either the Bordiga or the Palatini scroll.


title:
Bornes effectives pour la torsion des courbes elliptiques sur les corps
  des nombres

abstract:
  We give an effective form of the theorem of Mazur-Kamienny-Merel on the
torsion of elliptic curves over number fields.


title: Rationality of almost simple algebraic groups

abstract:
  We prove the stable rationality of almost simple algebraic groups, the
connected components of the Dynkin diagram of anisotropic kernel of which
contain at most two vertices. The (stable) rationality of many isotropic almost
simple groups with small anisotropic kernel and some related results over
$p$-adic and arbitrary fields are discussed.


title: The Hodge filtration on nonabelian cohomology

abstract:
  This is partly a survey article on nonabelian Hodge theory, but we also give
proofs of results that have only been announced elsewhere. In the introduction
we discuss a wide range of recent work on this subject and give some
references. In the body of the paper, we discuss Corlette's nonabelian Hodge
theorem, Hitchin's quaternionic structure on the moduli space of
representations of $\pi _1(X)$ (for a compact K\"ahler manifold $X$), and
Deligne's construction of the resulting twistor space. We then mention the
interpretation of Deligne's space of $\lambda$-connections as the Hodge
filtration of the nonabelian de Rham cohomology $M_{DR}(X,G)= H^1(X,G)$. We go
on to prove several properties of this Hodge filtration, such as Griffiths
transversality and regularity of the Gauss-Manin connection; a local triviality
property coming from Goldman and Millson's analysis of the singularities; and a
weight property coming from Langton's theory. We construct a compactification
of $M_{DR}$ using the Hodge filtration with its weight property. At the end, we
define a nonabelian version of the Noether-Lefschetz locus, and prove that it
is algebraic if the base of the fibration is compact. We pose the open problem
of studying degenerations of nonabelian Hodge structure sufficiently well to be
able to prove algebraicity of the Noether-Lefshetz locus even when the base is
quasiprojective.


title: On the constellations of Weierstrass points

abstract:
  We prove that the constellation of Weierstrass points characterizes the
isomorphism-class of double covering of curves of genus large enough.


title: On degree of polynomial mappings of $C^2$ to $C^2$

abstract: We prove two results about degree of polynomial mappings of $C^2$ to $C^2$.


title: On a result of Imin Chen

abstract:
  We give another proof of Imin Chen's result that the jacobian of the modular
curve X(p)_{non-split}, for p a prime number, is isogeneous to the new part of
the jacobian of X_0(p^2), using only the representation theory of the group
GL_2(Z/pZ). In fact, we prove a generalization of Chen's result for objects
with an action by GL_2(Z/pZ) in any pseudo-abelian Q-linear category.


title: Arf Rings and Characters

abstract:
  Algebraic curve branches can be classified according to their multiplicity
sequences. Arf's solution to this problem using Arf closures and possible
implementations of Henselization are discussed.


title: A connectivity lemma for the Albanese map

abstract:
  I prove that for any complex projective variety $X$ and a sufficiently large
integer $N$ all the fibers of Albanese map of the $N$-th configuration space of
$X$ are dominated by smooth connected projective varieties with vanishing ${\rm
H}^1$.


title:
On top Chern classes of universal bundles on moduli spaces of rank two
  coherent sheaves on the projective plane, or How to calculate the correlation
  function in SYM N=2 Nf=4 quantum field theory on complex projective plane

abstract:
  We explain how to calculate the correlation function for SYM N=2 SO(3)-gauge
QFT with 4 flavors in terms of top Chern classes of the universal bundles over
the moduli spaces of rank 2 stable torsion free coherent sheaves with det=-1 on
the complex projective plane. We give a direct geometrical description of these
moduli spaces, study a 2-dimensional torus action on them, and calculate
several initial coefficients of the correlation function via Bott residue
formula.


title: Elementary counterexamples to Kodaira vanishing in prime characteristic

abstract:
  We give elementary counterexamples to Kodaira vanishing in prime
characteristic constructed as the projectivization of Frobenius pull-backs of
natural bundles on the incidence variety of points and planes in projective
n-space (n>2).


title: On Classification of the extremal contraction from a smooth fourfold

abstract:
  We classify extremal divisorial contraction which contracts a divisor to a
curve from a smooth fourfold. We prove the exceptional divisor is $\Bbb
P^2$bundle or quadric bundle over a smooth curve and the contraction is the
blowing up along the curve.


title: A Lie algebra attached to a projective variety

abstract:
  Each choice of a K\"ahler class on a compact complex manifold defines an
action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty
set of such K\"ahler classes is given, then we prove that the corresponding
$\slt$-copies generate a semisimple Lie algebra. We investigate the formal
properties of the resulting representation and we work things out explicitly in
the case of complex tori, hyperk\"ahler manifolds and flag varieties. We pay
special attention to the cases where this leads to a Jordan algebra structure
or a graded Frobenius algebra.


title: Symmetries of Surface Singularities

abstract:
  The automorphism group ${\rm Aut}\: X$ of a weighted homogeneous normal
surface singularity $X$ has a maximal reductive algebraic subgroup $G$ which
contains every reductive algebraic subgroup of ${\rm Aut}\: X$ up to
conjugation. In all cases except the cyclic quotient singularities the
connected component $G_1$ of the unit equals ${\Bbb C}^*$. The induced action
of $G$ on the minimal good resolution of $X$ embeds the finite group $G/G_1$
into the automorphism group of the central curve $E_0$ of the exceptional
divisor. We describe $G/G_1$ as a subgroup of ${\rm Aut}\: E_0$ in case $E_0$
is rational as well as for simple elliptic singularities. Moreover, sufficient
conditions for $G$ to be a direct product $G_1 \times G/G_1$ are presented.
Finally, it is shown that $G/G_1$ acts faithfully on the integral homology of
the link of $X$.


title: Floppy Curves with Applications to Real Algebraic Curves

abstract:
  We show how one may sometimes perform singular ambient surgery on the complex
locus of a real algebraic curve and obtain what we call a floppy curve. A
floppy curve is a certain kind of singular surface in CP(2), more general than
the complex locus of a real nodal curve. We derive analogs for floppy curves of
known restrictions on real nodal curves. In particular we derive analogs of
Fielder's congruence for certain nonsingular curves and Viro's inequalities for
nodal curves which generalize those of Arnold and Petrovskii for nonsingular
curves. We also obtain a determinant condition for certain curves which are
extremal with respect to some of these equalities. One may prohibit certain
schemes for real algebraic curves by prohibiting the floppy curves which result
from singular ambient surgery. In this way, we give a new proof of Shustin's
prohibition of the scheme $1<2\coprod 1<18>>$ for a real algebraic curve of
degree eight.


title: Theta line bundles and the determinant of the Hodge bundle

abstract:
  In this note we examine the question of expressing the determinant of the
push forward of a symmetric line bundle on an abelian fibration in terms of the
pull back of the relative dualizing sheaf via the zero section.


title: Eisenstein series and quantum affine algebras

abstract:
  Let X be a smooth projectibe curve over a finite field. We consider the Hall
algebra H whose basis is formed by isomorphism classes of coherent sheaves on X
and whose typical structure constant is the number of subsheaves in a given
sheaf belonging to a given isomorphism class and with given isomorphism class
of quotient. We study this algebra in detail. It turns out that its structure
is strikingly similar to that of quantum affine algebras in their loop
realization of Drinfeld. In this analogy the set of simple roots of a
semisimple Lie algebra g corresponds to the set of cusp forms on X, the entries
of the Cartan matrix of g to the values of Rankin-Selberg L-functions
associated to two cusp forms. For the simplest instances of both theories (the
curve P^1, the Lie algebra g=sl_2) the analogy is in fact an identity.


title:
Recursive Formulas for the Characteristic Numbers of Rational Plane
  Curves

abstract:
  We derive recursive equations for the characteristic numbers of rational
nodal plane curves with at most one cusp, subject to point conditions, tangent
conditions and flag conditions, developing techniques akin to quantum
cohomology on a moduli space of stable lifts.


title:
Commutative Algebras of Ordinary Differential Operators with Matrix
  Coefficients

abstract:
  A classification of commutative integral domains consisting of ordinary
differential operators with matrix coefficients is established in terms of
morphisms between algebraic curves.


title:
The Symmetric Function h^0(\bar{M}_{0,n}, L_1^{x_1} \tensor...\tensor
  L_n^{x_n})

abstract:
  Let \bar{M}_{0,n} be the moduli space of pointed, genus 0 curves. Let L_i
denote the line bundle on \bar{M}_{0,n} associated to the i-th marked point
(the fiber of L_i is the cotangent space of the pointed curve at the i-th
point). Y_n=h^0(\bar{M}_{0,n}, L_1^{x_1} \tensor... \tensor L_n^{x_n}) is a
symmetric function of the variables x_1,... x_n. Let R be the ring of symmetric
functions in infinitely many variables. An explicit linear transformation T:
R-> R is found such that Y_n= T^{n-3} (1).


title: The Chow Ring of the Non-Linear Grassmannian

abstract:
  Let M_{P^k}(P^r, d) be the moduli space of unparameterized maps \mu:P^k ->
P^r satisfying \mu^*(O(1))= O(d). M_{P^k}(P^r,d) is a quasi-projective variety,
and, in case k=1, M_{P^1}(P^r,d) is the fundamental open cell of Kontsevich's
space of stable maps \bar{M}_{0,0}(P^r,d). It is shown that the Q-coefficient
Chow ring of M_{P^k}(P^r,d) is canonically isomorphic to the Chow ring of the
Grassmannian Gr(P^k, P^r)= M_{P^k}(P^r,1).


title:
On an unusual conjecture of Kontsevich and variants of Castelnuovo's
  lemma

abstract:
  Let $A=(a^i_j)$ be an orthogonal matrix with no entries zero. Let $B=(b^i_j)$
be the matrix defined by $b^i_j=\frac 1{a^i_j}$. M. Kontsevich conjectured that
the rank of $B$ is never equal to three. We interpret this conjecture
geometrically and prove it. The geometric statment can be understood as a
generalization of the Castelnouvo lemma and Brianchon's theorem.


title:
A high fibered power of a family of varieties of general type dominates
  a variety of general type

abstract:
  We prove the following theorem:
  Fibered Power Theorem: Let $X\rar B$ be a smooth family of positive
dimensional varieties of general type, with $B$ irreducible. Then there exists
an integer $n>0$, a positive dimensional variety of general type $W_n$, and a
dominant rational map $X^n_B \das W_n$.


title: Vector Bundles and Arithmetical Groups I. The higher Bruhat-Tits tree

abstract:
  We define and study a simplicial complex which is a homogeneous space for the
group $PGL(2, K)$ over a two-dimensional local field $K$. The complex is a
generalization of the tree studied by F. Bruhat, J. Tits, J.-P. Serre and P.
Cartier in the 60's and early 70's. Such complex can be canonically attached to
the triples $x \in C \subset X$ where $X$ is an algebraic surface, $C$ is an
irreducible curve and $x$ is a smooth point on $C$ and $X$. This construction
can be used for a description of the isomorphism set of vector bundles on $X$.


title:
On Extremal Contractions from Threefolds to Surfaces: the Case of One
  non-Gorenstein Point

abstract:
  Let $f\colon X\to S$ be an extremal contraction from a threefold with only
terminal singularities to a surface. We study local analytic structure such
contractions near degenerate fiber $C$ in the case when $C$ is irreducible and
$X$ has on $C$ only one non-Gorenstein point.


title: On the existence of good divisors on Fano varieties of coindex 3

abstract:
  We prove that the linear system $|-1/3K_X| on a non-singular Fano fivefold
$X$ of index 3 contains an irreducible divisor with only canonical
singularities.


title: Rank-2 ample vector bundles on some smooth rational surfaces

abstract:
  Some classification results for ample vector bundles of rank 2 on Hirzebruch
surfaces, and on Del Pezzo surfaces, are obtained. In particular, we classify
rank-2 ample vector bundles with $c_2$ less than 7 on Hirzebruch surfaces, and
with $c_2$ less than 4 on Del Pezzo surfaces.


title:
Double coset construction of moduli space of holomorphic bundles and
  Hitchin systems

abstract:
  We present a description of the moduli space of holomorphic vector bundles
over Riemann curves as a double coset space which is differ from the standard
loop group construction. Our approach is based on equivalent definitions of
holomorphic bundles, based on the transition maps or on the first order
differential operators. Using this approach we present two independent
derivations of the Hitchin integrable systems. We define a "superfree" upstairs
systems from which Hitchin systems are obtained by three step hamiltonian
reductions. A special attention is being given on the Schottky parameterization
of curves.


title: Donaldson and Seiberg-Witten invariants of algebraic surfaces

abstract:
  This paper, yet another account of Seiberg-Witten theory and its consequences
for algebraic surfaces, is the written version of the author's talk at the
Santa Cruz conference.


title:
Curves in the complement of a smooth plane cubic whose normalizations
  are $\Bbb A^1$

abstract:
  For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the
normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$
(for $d=7$ under some assumption). We also count plane rational quartic curves
intersecting $B$ at only one point.


title: On Genera of Smooth Curves in Higher Dimensional Varieties

abstract:
  We prove that for any smooth projective variety $X$ of dimension $\geq 3$,
there exists an integer $g_0=g_0(X)$, such that for any integer $g \geq g_0$,
there exists a smooth curve $C$ in $X$ with $g(C)=g$.


title:
A construction of separated quotient space and boundaries of symmetric
  spaces

abstract:
  We propose a construction of separated quotient space and apply this
construction for simple description of Study-Semple-Satake-Furstenberg-De
Concini-Procesi-Oshima boundary of symmetric space


title:
Sixth Painlev\'e Equation, Universal Elliptic Curve, and Mirror of
  $\bold{P}^2$

abstract:
  An algebro-geometric setting for the study of the Painlev\'e VI equation is
introduced. Hamiltonian form of the equation is realized on a twisted relative
cotangent bundle to the universal elliptic curve with labelled points of order
two. Relations with the theory of elliptic functions and the quantum cohomology
of projective plane are discussed.


title: Cycles on the Moduli Space of Abelian Varieties

abstract:
  In this paper a number of results on cycles on the moduli space of
principally polarized abelian varieties is presented. Results include a
determination of the tautological ring, bounds on the order of torsion of the
top Chern class $\lambda_g$ and a determination of the cycle classes of the
Ekedahl-Oort stratification in characteristic $p$. It includes as special cases
formulas for the classes of loci like the $p$-rank $\leq f$ locus or the
$a$-number $\geq a$-locus. The results on the tautological ring are my own
work, the results on the torsion of $\lambda_g$ and on the cycle classes of the
Ekedahl-Oort stratification are joint work with Torsten Ekedahl and some of the
results on curves are joint work with Carel Faber.


title: Complete Intersections K-Theory and Chern Classes

abstract:
  Throughout this abstruct $A$ will denote a noetherian commutative ring of
dimension $n$. The paper has two parts. Among the interesting results in Part-1
are the following: 1) {\it suppose that $f_1, f_2, ..., f_r$ (with $r \leq n$)
is a regular sequence in $A$ and suppose $Q$ is a projective $A$-module of rank
$r$ that maps onto the ideal $(f_1, f_2, ..., f_{r-1},f_r^{(r-1)!})$. Then
$[Q]=[Q_0 \oplus A]$ in $K_0(A)$ for some projective $A-module~Q_0$ of rank
$r-1$.} 2) The set $$F_0K_0(A) = \{[A/I] \in K_0(A): I~ is~ a~ locally~
complete ~intersection~ ideal~ in~ A~ of~ height~n \}$$ is a {\it subgroup} of
$K_0(A)$. We also show that if $A$ is a reduced affine algebra over a field $k$
then $F_0K_0(A)$ {\it is indeed the Zero Cycle Subgroup of} $K_0(A)$ {\it that
is generated by smooth maximal ideals} $\Cal M$ {\it of height} $n$. 3){\it let
$A$ be such that whenever $I$ is a locally complete intersection ideal of
height $n$ with $[A/I]=0$ then $I$ is the image of a projective $A-module$ of
rank $n$. Then for any locally complete intersection ideal $J$ of height $n$
with $[A/J]$ divisible by $(n-1)!$ in $F_0K_0(A)$, there is a projective
$A-module$ of rank $n$ that maps onto $J$}. The main result in Part-2 is the
following construction: 1) {\it let $X=Spec A$ be a Cohen-Macaulay scheme of
dimension $n$ and let $r_0,~r$ be two integers with $n/2 \leq r_0 \leq r \leq
n$. Let i) $Q_0$ be a projective $A-module$ of rank $r_0-1$ such that the
restriction $Q_0|Y$ is trivial for all locally complete intersection
subvarieties $Y$ of codimension at least $r_0$. Also ii) for $k= r_0$ to $r$,
let $I_k$ be locally complete intersection ideals of height $k$ so that
$I_k/I_k^2$ has a generators of the type $f_1, ..., f_{k-1}, f_k^{(k-1)!}$.
Then there is a projective $A-module~Q$ of rank $r$ such that


title: On Contractions of smooth varieties

abstract:
  Let $\f: X \ra Z$ be a proper surjective map from a smooth complex manifold
$X$ onto a normal variety $Z$. If $\f$ has connected fibers and $-K_X$ is
$\f$-ample then $\f$ is called a good contraction. In the present paper we
study good contractions, fibers of which have dimension less or equal than two:
after describing possible two dimensional isolated fibers we discuss their
scheme theoretic structure and the geometry of $\f:X\ra Z$ nearby such a fiber.
If $dimX=4$ and $\f$ is birational with an isolated 2 dimensional fiber then we
obtain a complete description of $\f$. We provide also a description of a 4
dimensional conic fibration with an isolated fiber which is either a plane or a
quadric. We construct pertinent examples.


title:
Symmetric polynomials and divided differences in formulas of
  intersection theory

abstract:
  The goal of the paper is two-fold. At first, we attempt to give a survey of
some recent applications of symmetric polynomials and divided differences to
intersection theory. We discuss: polynomials universally supported on
degeneracy loci; some explicit formulas for the Chern and Segre classes of
Schur bundles with applications to enumerative geometry; flag degeneracy loci;
fundamental classes, diagonals and Gysin maps; intersection rings of G/P and
formulas for isotropic degeneracy loci; numerically positive polynomials for
ample vector bundles.
  Apart of surveyed results, the paper contains also some new results as well
as some new proofs of earlier ones: how to compute the fundamental class of a
subvariety from the class of the diagonal of the ambient space; how to compute
the class of the relative diagonal using Gysin maps; a new formula for pushing
forward Schur's Q- polynomials in Grassmannian bundles; a new formula for the
total Chern class of a Schur bundle; another proof of Schubert's and
Giambelli's enumeration of complete quadrics; an operator proof of the
Jacobi-Trudi formula; a Schur complex proof of the Giambelli-Thom-Porteous
formula.


title: Descent from the form ring and Buchsbaum rings

abstract:
  There is a spectral sequence technique in order to estimate the local
cohomology of a ring by the local cohomology of a certain form ring. As
applications there are information on the descent of homological properties
(Cohen-Macaulay, Buchsbaum etc.) from the form ring to the ring itself. In the
case of Buchsbaum ring there is a discussion of the descent of the surjectivity
of a natural map into the local cohomology.


title: On the period map for abelian covers of algebraic varieties

abstract:
  We show that infinitesimal Torelli for $n$-forms holds for abelian covers of
algebraic varieties of dimension $n\ge 2$, under some explicit ampleness
assumptions on the building data of the cover. Moreover, we prove a variational
Torelli result for some families of abelian covers.


title: Height Zeta Functions of Toric Varieties

abstract:
  We investigate analytic properties of height zeta functions of toric
varieties. Using the height zeta functions, we prove an asymptotic formula for
the number of rational points of bounded height with respect to an arbitrary
line bundle whose first Chern class is contained in the interior of the cone of
effective divisors


title: Algebraically constructible functions

abstract:
  An algebraic version of Kashiwara and Schapira's calculus of constructible
functions is used to describe local topological properties of real algebraic
sets, including Akbulut and King's numerical conditions for a stratified set of
dimension three to be algebraic. These properties, which include
generalizations of the invariants modulo 4, 8, and 16 of Coste and Kurdyka, are
defined using the link operator on the ring of constructible functions.


title: Nonrational Covers of $CP^m\times CP^n$

abstract:
  The aim of the paper is to construct examples of cyclic covers of $CP^m\times
CP^n$ which are rationally connected but are not rational. The crucial point
turns out to be the study of Del Pezzo surfaces over function fields in
positive characteristic.


title: Equivalences of derived categories and K3 surfaces

abstract:
  We consider derived categories of coherent sheaves on smooth projective
varieties. We prove that any equivalence between them can be represented by an
object on the product. Using this, we give a necessary and sufficient condition
for equivalence of derived categories of two K3 surfaces.


title: Algebraic cycles and topology of real Enriques surfaces

abstract:
  For a real Enriques surface Y we prove that every homology class in H_1(Y(R),
Z/2) can be represented by a real algebraic curve if and only if all connected
components of Y(R) are orientable. Furthermore, we give a characterization of
real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we
determine the Brauer group of any real Enriques surface.


title: Finite sets in fibres of holomorphic maps

abstract:
  We consider the maximal number of arbitrary points in a special fibre that
can be simultaneously approached by points in one sequence of general fibres.
Several results about this topological invariant and their applications
describe the structure of holomorphic maps. In particular, we get a lower bound
on the number of points in the general fibre of a generically finite map.


title: The algebraic formalism of soliton equations over arbitrary base fields

abstract:
  The aim of this paper is to offer an algebraic construction of
infinite-dimensional Grassmannians and determinant bundles (and therefore valid
for arbitrary base fields). As an application we construct the $\tau$-function
and formal Baker-Akhiezer functions over arbitrary fields, by proving the
existence of a ``formal geometry'' of local curves analogous to the geometry of
global algebraic curves.
  We begin by defining the functor of points, $\fu{\gr}(V,V^+)$, of the
Grassmannian of a $k$-vector space $V$ in such a way that its rational points
are precisely the points of the Grassmannian defined by Segal-Wilson, although
the points over an arbitrary $k$-scheme $S$ have been not previously
considered. This definition of the functor $\fu{\gr}(V,V^+)$ allows us to prove
that it is representable by a separated $k$-scheme $\gr(V,V^+)$. Using the
theory of determinants of Knudsen and Mumford, the determinant bundle is
constructed. This is one of the main results of the paper because it implies
that we can define ``infinite determinants'' in a completely algebraic way.


title: Descent of Deligne groupoids

abstract:
  To any non-negatively graded dg Lie algebra $g$ over a field $k$ of
characteristic zero we assign a functor $\Sigma_g: art/k \to Kan$ from the
category of commutative local artinian $k$-algebras with the residue field $k$
to the category of Kan simplicial sets. There is a natural homotopy equivalence
between $\Sigma_g$ and the Deligne groupoid corresponding to $g$.
  The main result of the paper claims that the functor $\Sigma$ commutes up to
homotopy with the "total space" functors which assign a dg Lie algebra to a
cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial
set. This proves a conjecture of Schechtman which implies that if a deformation
problem is described ``locally'' by a sheaf of dg Lie algebras $g$ on a
topological space $X$ then the global deformation problem is described by the
homotopy Lie algebra $R\Gamma(X,g)$.