title: Deformations of cones over hyperelliptic curves

abstract:
  We determine the versal deformation of cones, in the simplest case: cones
over hyperelliptic curves of high degree. In particular, we show that for
degree $4g+4$, the highest degree for which interesting deformations exist, the
number of smoothing components is $2^{2g+1}$ ($g\neq3$).
  We review in a general setting the relation of $T^1(-1)$ with Wahl's Gaussian
map. We prove that $T^1(-1)$ vanishes for a general curve and an arbitrary
embedding line bundle of degree at least $2g+11$. To find $T^2$ for
hyperelliptic cones with the Main Lemma of [Behnke--Christophersen], we compute
$T^1$ for the cone over $d$ points on a rational normal curve of degree
$d-g-1$, using explicit equations. Actually, the equations for the cone over a
hyperelliptic curve have a nice structure. We give an interpretation of
$T^2(-2)$ in terms of this structure.
  Smoothing components are related to surfaces with $C$ as hyperplane section.
An explicit description of the corresponding infinitesimal deformations allowss
to conclude that the base space is a complete intersection of degree
$2^{2g+1}$. We also consider smoothing data in the sense of [Looijenga--Wahl].


title: Theta Functions for $\SL(n)$ versus $\GL(n)$

abstract:
  Over a smooth complex projective curve $C$ of genus $g$ let $\M (n,d)$ be the
moduli space of semistable bundles of rank $n$ and degree $d$ on $C$, and $\SM
(n,L)$, the moduli space of those bundles whose determinant is isomorphic to a
fixed line bundle $L$ over $C$. Let $\theta_F$ and $\theta$ be theta bundles
over these two moduli spaces. We prove a simple formula relating their spaces
of sections: if $h=\gcd (n,d)$ is the greatest common divisor of $n$ and $d$,
and $L\in \Pic ^d(C)$, then $$\dim H^0(\SM (n,L), \theta^k) \cdot k^g=\dim
H^0(\M(n,d),\theta_F^k)\cdot h^g.$$ We also formulate a conjectural duality
between these two types of spaces of sections.


title: Free pencils on divisors

abstract:
  Let X be a smooth projective variety defined over an algebraically closed
field, and let Y in X be a reduced and irreducible ample divisor in X. We give
a numerical sufficient condition for a base point free pencil on $Y$ to be the
restriction of a base point free pencil on $X$. This result is then extended to
families of pencils and to morphisms to arbitrary smooth curves. Serrano had
already studied this problem in the case n=2 and 3, and Reider had then
attacked it in the case $n=2$ using vector bundle methods based on Bogomolov's
instability theorem on a surface (char(k)=0). The argument given here is based
on Bogomolov's theorem on an n-dimensional variety, and on its recent
adaptations to the setting of prime charachterstic (due to Shepherd-Barron and
Moriwaki).


title:
Seshadri constants, gonality of space curves and restriction of stable
  bundles

abstract:
  We define the Seshadri constant of a space curve and consider ways to
estimate it. We then show that it governs the gonality of the curve. We use an
argument based on Bogomolov's instability theorem on a threefold. The same
methods are then applied to the study of the behaviour of a stable vector
bundle on P^3 under restriction to curves and surfaces.


title: The Variety of Positive Superdivisors of a Supercurve (Supervortices)

abstract:
  The supersymmetric product of a supercurve is constructed with the aid of a
theorem of algebraic invariants and the notion of positive relative
superdivisor (supervortex) is introduced. A supercurve of positive
superdivisors of degree 1 (supervortices of vortex number 1) of the original
supercurve is constructed as its supercurve of conjugate fermions, as well as
the supervariety of relative positive superdivisors of degre p (supervortices
of vortex number p.) A universal superdivisor is defined and it is proved that
every positive relative superdivisor can be obtained in a unique way as a
pull-back of the universal superdivisor. The case of SUSY-curves is discussed.


title: Birational Equivalences of Vortex Moduli

abstract:
  We construct a finite dimensional Kaehler manifold with a holomorphic,
symplectic circle action whose symplectic reduced spaces may be identified with
the tau-vortex moduli spaces (or tau-stable pairs). The Morse theory of the
circle action induces natural birational maps between the reduced spaces for
different values of tau which in the case of rank two bundles can be
canonically resolved in a sequence of blow-ups and blow-downs.


title:
Refined intersection products and limiting linear subspaces of
  hypersurfaces

abstract:
  Let $X$ be a hypersurface of degree $d$ in $\Bbb P^n$ and $F_X$ be the scheme
of $\Bbb P^r$'s contained in $X$. If $X$ is generic, then $F_X$ will have the
expected dimension (or empty) and its class in the Chow ring of $G(r+1,n+1)$ is
given by the top Chern class of the vector bundle $S^dU^*$, where $U$ is the
universal subbundle on the Grassmannian $G(r+1,n+1)$. When we deform a generic
$X$ into a degenerate $X_0$, the dimension of $F_X$ can jump. In this case,
there is a subscheme $F_{lim}$ of $F_{X_0}$ with the expected dimension which
consists of limiting $\Bbb P^r$'s in $X_0$ with respect to a general
deformation. The simplest example is the well-known case of $27$ lines in a
generic cubic surface. If we degenerate the surface into the union of a plane
and a quadric, then there are infinitely many lines in the union. Which $27$
lines are the limiting ones and how many of them are in the plane and how many
of them are in the quadric? The goal of this paper is to study $F_{lim}$ in
general.


title: What can be computed in algebraic geometry?

abstract:
  This paper is a survey of computational issues in algebraic geometry, with
particular attention to the theory of Grobner bases and the regularity of an
algebraic variety.
  1. A geometric introduction to Grobner bases.
  2. An algebraic introduction to Grobner bases.
  3. Bounds in algebraic geometry, and regularity and complexity questions.
  4. Applications.


title:
Holomorphic Slices, Symplectic Reduction and Multiplicities of
  Representations

abstract:
  I prove the existence of slices for an action of a reductive complex Lie
group on a K\"ahler manifold at certain orbits, namely those orbits that
intersect the zero level set of a momentum map for the action of a compact real
form of the group. I give applications of this result to symplectic reduction
and geometric quantization at singular levels of the momentum map. In
particular, I obtain a formula for the multiplicities of the irreducible
representations occurring in the quantization in terms of symplectic invariants
of reduced spaces, generalizing a result of Guillemin and Sternberg.


title:
Schur quadrics, cubic surfaces and rank 2 vector bundles over the
  projective plane

abstract:
  A cubic surface in $P^3$ is known to contain 27 lines, out of which one can
form 36 Schlafli double - sixes i.e., collections $l_1,...,l_6, l'_1,...,
l'_6\}$ of 12 lines such that each $l_i$ meets only $l'_j, j\neq i$ and does
not meet $l_j, j\neq i$. In 1881 F. Schur proved that any double - six gives
rise to a certain quadric $Q$ , called Schur quadric which is characterized as
follows: for any $i$ the lines $l_i$ and $l'_i$ are orthogonal with respect to
(the quadratic form defining) $Q$. The aim of the paper is to relate Schur's
construction to the theory of vector bundles on $P^2$ and to generalize this
construction along the lines of the said theory.


title: Recovering of curves with involution by extended Prym data

abstract:
  With every smooth, projective algebraic curve $\tilde{C}$ with involution
$\sigma :\tilde{C}\longrightarrow \tilde{C}$ without fixed points is associated
the Prym data which consists of the Prym variety $P:=(1-\sigma )J(\tilde{C})$
with principal polarization $\Xi$ such that $2\Xi$ is algebraically equivalent
to the restriction on $P$ of the canonical polarization $\Theta $ of the
Jacobian $J(\tilde{C})$. In contrast to the classical Torelli theorem the Prym
data does not always determine uniquely the pair $(\tilde{C},\sigma )$ up to
isomorphism. In this paper we introduce an extension of the Prym data as
follows. We consider all symmetric theta divisors $\Theta $ of $J(\tilde{C})$
which have even multiplicity at every point of order 2 of $P$. It turns out
that they form three $P_2$ orbits. The restrictions on $P$ of the divisors of
one of the orbits form the orbit $\{ 2\Xi \} $, where $\Xi $ are the symmetric
theta divisors of $P$. The other restrictions form two $P_2$-orbits
$O_1,O_2\subset \mid 2\Xi \mid $. The extended Prym data consists of $(P,\Xi )$
together with $O_1,O_2$. We prove that it determines uniquely the pair
$(\tilde{C} ,\sigma )$ up to isomorphism provided $g(\tilde{C})\geq 3$. The
proof is analogous to Andreotti's proof of Torelli's theorem and uses the Gauss
map for the divisors of $O_1,O_2$. The result is an analog in genus $>1$ of a
classical theorem for elliptic curves.


title: Compactifications of moduli spaces inspired by mirror symmetry

abstract:
  We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds,
using the one-loop semiclassical approximation. The data being parameterized
includes a choice of complex structure on the manifold, as well as some ``extra
structure'' described by means of classes in H^2. The expectation that this
moduli space is well-behaved in these ``extra structure'' directions leads us
to formulate a simple and compelling conjecture about the action of the
automorphism group on the K\"ahler cone. If true, it allows one to apply
Looijenga's ``semi-toric'' technique to construct a partial compactification of
the moduli space. We explore the implications which this construction has
concerning the properties of the moduli space of complex structures on a
``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a
similarity which might have been noticed between certain work of Mumford and of
Mori from the 1970's produces (with hindsight) evidence for mirror symmetry
which was available in 1979. [The author is willing to mail hardcopy preprints
upon request.]


title: A General Noether-Lefschetz Theorem and applications

abstract:
  In this paper we generalize the classical Noether-Lefschetz Theorem to
arbitrary smooth projective threefolds. Let $X$ be a smooth projective
threefold over complex numbers, $L$ a very ample line bundle on $X$. Then we
prove that there is a positive integer $n_0(X,L)$ such that for $n \geq
n_0(X,L)$, the Noether-Lefschetz locus of the linear system $H^0(X,L^n)$ is a
countable union of proper closed subvarieties of $\P(H^0(X,L^n)^*)$ of
codimension at least two. In particular, the {\em general singular member} of
the linear system $H^0(X,L^n)$ is not contained in the Noether-Lefschetz locus.
  As an application of our main theorem we prove the following result: Let $X$
be a smooth projective threefold, $L$ a very ample line bundle. Assume that $n$
is very large. Let $S=\P(H^0(X,L^n)^*)$, let $K$ denote the function field of
$S$. Let ${\cal Y}_K$ be the generic hypersurface corresponding to the sections
of $H^0(X,L^n)$. Then we show that the natural map on codimension two cycles
  $$ CH^2(X_{\C}) \to CH^({\cal Y}_K) $$ is injective. This is a weaker version
of a conjecture of M. V. Nori, which generalises the Noether-Lefschetz theorem
on codimension one cycles on a smooth projective threefolds to arbitrary
codimension


title: A Finiteness Theorem for Elliptic Calabi-Yau Threefolds

abstract:
  We prove that up to birational equivalence, there exists only a finite number
of families of Calabi-Yau threefolds (i.e. a threefold with trivial canonical
class and factorial terminal singularities) which have an elliptic fibration to
a rational surface. This strengthens a result of B. Hunt that there are only a
finite number of possible Euler characteristics for such threefolds.


title: Log contractions and equidimensional models of elliptic threefolds

abstract:
  This work was initially motivated by Miranda's work on elliptic Weierstrass
threefolds. Miranda [Mi] describes a smooth equidimensional (flat) model for
any elliptic Weierstrass threefold; such models occur naturally in the study of
moduli spaces. In this paper we use minimal model theory to link birational
maps of log surfaces (log contractions) to equidimensional fibrations of
elliptic threefolds. As a corollary we show that an elliptic fibration of
positive Kodaira dimension has a minimal model with an equidimensional
birationally equivalent elliptic fibration satisfying certain good properties
that naturally generalize the case of elliptic fibrations of surfaces. We apply
the results to the case of Weierstrass models and give a global explanation of
Miranda's algorithm.


title: Kodaira energies of polarized log varieties

abstract:
  Let L be an ample line bundle on a log variety (V, D) having only log
terminal singularities.The Kodaira energy of such a triple (V, D, L) is defined
as follows: \kappa\epsilon=-Inf{t\in Q | K(V,D)+tL is big}. Here K(V,D)=K_V+D
is the log canonical bundle of (V,D) and "big" means \kappa=dim V. We first
show that the rationality of \kappa\epsilon, when it is negative, follows from
log Flip theorems (thus it is proved for dim V\le 3 at present). Second we
consider the case in which D=0, dim V=3 and V has only terminal singularities.
Let S be the set of all the possible values of Kodaira energies of such
polarized log threefolds. We prove that S has no negative limit point. This
result is closely related to the boundedness of Fano 3-folds.


title: Inequality of Bogomolov-Gieseker's type on arithmetic surfaces

abstract:
  Let K be an algebraic number field, O_K the ring of integers of K, and f : X
--> Spec(O_K) an arithmetic surface. Let (E, h) be a rank r Hermitian vector
bundle on X such that $E$ is semistable on the geometric generic fiber of f. In
this paper, we will prove an arithmetic analogy of Bogomolov-Gieseker's
inequality: c_2(E, h) - (r-1)/(2r) c_1(E, h)^2 >= 0.


title: The Noether exponent and Jacobi formula

abstract:
  For any polynomial mapping $F=(F_1,\dots ,F_n)$ of $\Cal C^n$ with a finite
number of zeros we define the Noether exponent $\nu(F)$. We prove the Jacobi
formula for all polynomials of degree strictly less than $\sum_{i=1}^n (\deg
F_i-1)-\nu(F)$.


title: An inequality for polynomial mappings

abstract: We give an estimate of the growth of a polynonial mapping of $C^n$.


title: Injectivity on one line

abstract:
  Let $k$ be an algebraically closed field of characteristic zero. Let
$H:k^2\to k^2$ be a polynomial mapping such that the Jacobian $\text{Jac}\,H$
is a non-zero constant. In this note we prove, that if there is a line $l
\subset k^2$ such that $H|_l:l\to k^2$ is an injection, then $H$ is a
polynomial automorphism.


title: On the approximate roots of polynomials

abstract:
  We give a simplified approach to the Abhyankar--Moh theory of approximate
roots. Our considerations are based on properties of the intersection
multiplicity of local curves.


title: Finding Sparse Systems of Parameters

abstract:
  For several computational procedures such as finding radicals and Noether
normalizations, it is important to choose as sparse as possible a system of
parameters in a polynomial ideal or modulo a polynomial ideal. We describe new
strategies for these tasks, thus providing solutions to problems (1) and (2)
posed in [Eisenbud-Huneke-Vasconcelos 1992].


title: On the fundamental group of an abelian cover

abstract:
  We study the behaviour of the topological fundamental group under totally
ramified abelian covers (a special case of abelian Galois covers) of complex
projective varieties of dimension at least 2.


title: A Kaehler Structure on the Space of String World-Sheets

abstract:
  Let (M,g) be an oriented Lorentzian 4-manifold, and consider the space S of
oriented, unparameterized time-like 2-surfaces in M (string world-sheets) with
fixed boundary conditions. Then the infinite-dimensional manifold S carries a
natural complex structure and a compatible (positive-definite) Kaehler metric h
on S determined by the Lorentz metric g. Similar results are proved for other
dimensions and signatures, thus generalizing results of Brylinski regarding
knots in 3-manifolds. Generalizing the framework of Lempert, we also
investigate the precise sense in which S is an infinite-dimensional complex
manifold.


title: The automorphism group of the moduli space of semi stable vector bundles

abstract:
  Let ${\cal S}{\cal U}(r, L_0)$ denote the moduli space of semi stable vector
bundles of rank $r$ and fixed determinant $L_0$ of degree $d$ on a smooth curve
$C$ of genus $g \geq 3$. In this paper we describe the group of automorphisms
of $ {\cal S}{\cal U}(r, L_0) $. The analogue of this result is carried out for
the space ${\cal U}(r,d) $ of semi stable vector bundles of rark $r$ and degree
$d$. As an application of the technics we use, we give in the appendix at the
end of the paper a proof of the Torelli theorem for the moduli spaces ${\cal
S}{\cal U}(r, L_0) $ for any rank $r$ and degree $d$.


title: Picard groups of Hilbert schemes of curves

abstract:
  We calculate the Picard group, over the integers, of the Hilbert scheme of
smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in
${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the
classes of the generators in terms of some ``natural'' divisor classes.


title: Linear Structure on Calabi-Yau Moduli Spaces

abstract:
  We show that the formal moduli space of a Calabi-Yau manifold $X^n$ carries a
linear structure, as predicted by mirror symmetry. This linear structure is
canonically associated to a splitting of the Hodge filtration on $H^n(X)$.


title: Very ample linear systems on abelian varieties

abstract:
  Let $(X,L)$ be a polarized complex abelian variety of dimension $g$ where $L$
is a polarization of type $(1,...,1,d)$. For $(X,L)$ genberic we prove the
following:
  (1) If $d \ge g+2$, then $\phi_L\colon X \to {\bf P}^{d-1}$ defines a
birational morphism onto its image.
  (2) If $d > 2^g$, then $L$ is very ample.
  We show the latter by checking it on a suitable rank-$(g-1)$-degeneration.


title:
Gromov Invariants for Holomorphic Maps from Riemann Surfaces to
  Grassmannians

abstract:
  Two compactifications of the space of holomorphic maps of fixed degree from a
compact Riemann surface to a Grassmannian are studied. It is shown that the
Uhlenbeck compactification has the structure of a projective scheme and is
dominated by the algebraic compactification arising as a Grothendieck Quot
scheme. The latter may be embedded into the moduli space of solutions to a
generalized version of the vortex equations studied by Bradlow. This gives an
effective way of computing certain intersection numbers (known as Gromov
invariants) on the space of holomorphic maps into Grassmannians. We carry out
these computations in the case where the Riemann surface has genus one.


title: Bogomolov Instability and Kawamata-Viehweg Vanishing

abstract:
  The purpose of this note is to show how the Kawamata-Viehweg vanishing
theorem for fractional divisors leads to a quick new proof of Bogomolov's
instability theorem for rank two vector bundles on an algebraic surface.


title: Trisecant Lines And Jacobians, II

abstract:
  Let $\Theta$ be a symmetric theta divisor on an indecomposable principally
polarized complex abelian variety $X$. The linear system $|2\Theta |$ defines a
morphism $K:X\ra |2\Theta |^*$, whose image is the Kummer variety $K(X)$ of
$X$. When $(X,\theta)$ is the Jacobian of an algebraic curve, there are
infinitely many trisecants lines to $K(X)$. Welters has conjectured that the
existence of one trisecant line to the Kummer variety should characterize
Jacobians. The purpose of this article is to show the following weak version of
Welters conjecture: $X$ is a Jacobian if and only if there exist points $a,b,c$
of $X$ such that (i) the subgroup of $X$ generated by $a-b$ and $b-c$ is dense
in $X$, (ii) the points $K(a)$, $K(b)$ and $K(c)$ are distinct and collinear.
This improves on previous results obtained by the author (Trisecant Lines And
Jacobians, J. Alg. Geom. 1 (1992), 5--14). Various degenerate cases of the
conjecture are also considered.


title:
Distribution of Energy Levels of a Quantum Free Particle on a Surface of
  Revolution

abstract:
  We prove that the error term $\La(R)$ in the Weyl asymptotic formula $$\#\{
E_n\le R^2\}={\Vol M\over 4\pi} R^2+\La(R),$$ for the Laplace operator on a
surface of revolution $M$ satisfying a twist hypothesis, has the form $\La(R)
=R^{1/2}F(R)$ where $F(R)$ is an almost periodic function of the Besicovitch
class $B^2$, and the Fourier series of $F(R)$ in $B^2$ is $\sum_\g
A(\g)\cos(|\g|R-\phi)$ where the sum goes over all closed geodesics on $M$, and
$A(\g)$ is computed through simple geometric characteristics of $\g$. We extend
this result to surfaces of revolution, which violate the twist hypothesis and
satisfy a more general Diophantine hypothesis. In this case we prove that
$\La(R)=R^{2/3}\Phi(R)+R^{1/2}F(R)$, where $\Phi(R)$ is a finite sum of
periodic functions and $F(R)$ is an almost periodic function of the Besicovitch
class $B^2$. The Fourier series of $\Phi(R)$ and $F(R)$ are computed.


title: Classification of smooth congruences with a fundamental curve

abstract:
  We give a classification and a construction of all smooth $(n-1)$-dimensional
varieties of lines in ${\bf P}\sp n$ verifying that all their lines meet a
curve. This also gives a complete classification of $(n-1)$-scrolls over a
curve contained in $G(1,n)$.


title: On stability of tangent bundles of Fano manifolds with b_2=1

abstract:
  In the present paper we discuss stability of the tanget bundle of a Fano
n-fold of index >= n-2 and b_2=1. For example, we prove that all Fano 4-folds
with b_2=1 have stable tangent bundle. For this purpose we prove some vanishing
theorems for sheaves of twisted holomorphic forms on ample divisors and cyclic
coverings of manifolds having "special cohomology".


title: On the Hodge Structure of Projective Hypersurfaces in Toric Varieties

abstract:
  This paper generalizes classical results of Griffiths, Dolgachev and
Steenbrink on the cohomology of hypersurfaces in weighted projective spaces.
Given a $d$-dimensional projective simplicial toric variety $P$ and an ample
hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate
ring $S$ of $P$ (as defined in an earlier paper of the first author), we show
that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally
isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian
ideal of $f$. We then discuss how this relates to the primitive cohomology of
$X$. Also, if $T$ is the torus contained in $X$, then the intersection of $X$
and $T$ is an affine hypersurface in $T$, and we show how recent results of the
second author can be stated using various ideals in the ring $S$.
  To prove our results, we must give a careful description (in terms of $S$) of
$d$-forms and $(d-1)$-forms on the toric variety $P$. For completeness, we also
provide a proof of the Bott-Steenbrink-Danilov vanishing theorem for simplicial
toric varieties.
  Other topics considered in the paper include quasi-smooth hypersurfaces and
$V$-submanifolds, the structure of the complement of $U$ when $P$ is
represented as the quotient of an open subset $U$ of affine space, a
generalization of the Euler exact sequence on projective space, and the
relation between graded pieces of $R/J(f)$ and the moduli of ample
hypersurfaces in $P$.


title: Localization for nonabelian group actions

abstract:
  Suppose $X$ is a compact symplectic manifold acted on by a compact Lie group
$K$ (which may be nonabelian) in a Hamiltonian fashion, with moment map $\mu: X
\to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction $\xred = \mu^{-1}(0)/K$.
There is then a natural surjective map $\kappa_0$ from the equivariant
cohomology $H^*_K(X) $ of $X$ to the cohomology $H^*(\xred)$. In this paper we
prove a formula (Theorem 8.1, the residue formula) for the evaluation on the
fundamental class of $\xred$ of any $\eta_0 \in H^*(\xred)$ whose degree is the
dimension of $\xred$, provided that $0$ is a regular value of the moment map
$\mu$ on $X$. This formula is given in terms of any class $\eta \in H^*_K(X)$
for which $\kappa_0(\eta ) = \eta_0$, and involves the restriction of $\eta$ to
$K$-orbits $KF$ of components $F \subset X$ of the fixed point set of a chosen
maximal torus $T \subset K$. Since $\kappa_0$ is


title: Vector bundles and $SO(3)$ invariants for elliptic surfaces I

abstract:
  This paper is the first in a series of three devoted to the smooth
classification of simply connected elliptic surfaces. The method is to compute
some coefficients of Donaldson polynomials of $SO(3)$ invariants whose second
Stiefel-Whitney class is transverse to the unique primitive class $\kappa$ such
that a positive multiple of $\kappa$ is the class of a general fiber on the
surface. In this paper, we collect preliminary results on elliptic surfaces and
vector bundles and give the general outline of the argument.


title:
Vector bundles and $SO(3)$ invariants for elliptic surfaces II: The case
  of even fiber degree

abstract:
  This paper is the second in a series of three devoted to the smooth
classification of simply connected elliptic surfaces. In this paper, we study
the case where one of the multiple fibers has even multiplicity, and describe
the moduli space of stable rank two vector bundles with the appropriate first
Chern class needed to calculate Donaldson polynomials. The analysis is in many
ways parallel to the analysis in the case of vector bundles of trivial
determinant, but the asymmetry between the multiplicities also appears in the
moduli space. In this way, the first coefficient of an appropriate Donaldson
polynomial determines one of the multiplicities.


title: Arithmetic Bogomolov-Gieseker's inequality

abstract:
  Let f : X --> Spec(Z) be an arithmetic variety of dimension d >= 2 and (H, k)
an arithmetically ample Hermitian line bundle on X. Let (E, h) be a rank r
vector bundle on X. In this paper, we will prove that if E is semistable with
respect to H on each connected component of the infinite fiber of X, then
  { c_2(E, h) - (r-1)/(2r) c_1(E, h)^2 } c_1(H, k)^{d-2} >= 0. Moreover, if the
equality of the above inequality holds, then E is projectively flat and h is a
weakly Einstein-Hermitian metric.


title: Duistermaat-Heckman measures in a non-compact setting

abstract:
  We prove a \dh type formula in a suitable non-compact setting. We use this
formula to evaluate explicitly the pushforward of the Liouville measure via the
moment map of both an abelian and a non-abelian group action. As an application
we obtain the classical analogues of well-known multiplicity formulas for the
holomorphic discrete series representations.


title: Boundary of the Picard Group of a Singular Curve

abstract:
  In this work we explore the boundary of the jacobian of a singular curve in
the compactified picard group. We formulate a functor that helps to determine
this boundary and show that this functor is representable. We try to dertermine
the points on the boundary.


title: Compactified Jacobian of some Singular Curves

abstract:
  We look at the decomposition of the compactified jacobian of a singular curve
into components and discuss some examples.


title: Hyperholomorphic bundles

abstract:
  Hyperholomorphic bundle is a bundle with connection defined over a
hyperkaehler manifold such that this connection is holomorphic with respect to
all complex structures induced by a hyperkaehler structure. A hyperholomorphic
connection is Yang-Mills. If a stable bundle has first two Chern classes
invariant with respect to ${\Bbb H}$, then it admits a hyperholomorphic
connection. Deformation spaces of hyperholomorphic bundles are hyperkaehler.
There are no higher obstructions to a deformation besides Ioneda product. This
deformation space does not depend on a base manifold.


title: Hyperkaehler and holomorphic symplectic geometry

abstract:
  We prove that all complex analytic subvarieties of a generic compact
hyperkaehler manifold are even-dimensional. Moreover, these subvarieties are
holomorphically symplectic.


title:
Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau
  Complete Intersections in Toric Varieties

abstract:
  We formulate general conjectures about the relationship between the A-model
connection on the cohomology of a $d$-dimensional Calabi-Yau complete
intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety
${\bf P}_{\Sigma}$ and the system of differential operators annihilating the
special hypergeometric function $\Phi_0$ depending on the fan $\Sigma$. In this
context, the Mirror Symmetry phenomenon can be interpreted as the following
twofold characterization of the series $\Phi_0$. First, $\Phi_0$ is defined by
intersection numbers of rational curves in ${\bf P}_{\Sigma}$ with the
hypersurfaces $V_i$ and their toric degenerations. Second, $\Phi_0$ is the
power expansion near a boundary point of the moduli space of the monodromy
invariant period of the holomorphic differential $d$-form on an another
Calabi-Yau $d$-fold $V'$ which is called Mirror of $V$. Using the generalized
hypergeometric series, we propose a general construction for Mirrors $V'$ of
$V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.


title: On the Griffiths group of the cubic sevenfold

abstract:
  We prove that the Griffiths group of 3-cycles homologous to zero modulo
algebraic equivalence, on a generic hypersurfaces of dimension 7 and degree 3
is not finitely generated, even when tensored with Q. Using this and a result
of Nori, we give examples of varieties for which some Griffiths group is not
finitely generated (modulo torsion) but whose corresponding intermediate
Jacobian is trivial.


title: The Euler Series of Restricted Chow Varieties

abstract:
  Let X be an algebraic projective variety in {\bf P}^n. Denote by {\cal
C}_{\lambda} the space of all effective cycles on X whose homology class is
\lambda \in H_{2p} (X,{\bf Z}). It is easy to show that {\cal C}_{\lambda} is
an algebraic projective variety. Let \chi ({\cal C}_{\lambda} be its Euler
characteristic. Define the Euler series of X by
  E_{p} = \sum_{\lambda\in{C}} \, \chi({\cal C}_{\lambda} \lambda
  \in \, {\bf Z}[[C]]
  where {\bf Z}[[C]] is the full algebra over {\bf Z} of the monoid C of all
homology classes of effective p-cyles on X. This algebra is the ring of
function (with respect the convolution product) over C. Denote by {\bf Z}[C]
the ring of functions with finite support on C. We say that an element of {\bf
Z}[[C]] is rational if it is the quotient of two elements in {\bf Z}[C]. If a
basis for homology is fixed we can associated to any rationa element a rational
function and therefore compute the Euler characteristic of {\cal C}_{\lambda}.
We prove that E_p is rational for any projective variety endowed with an
algebraic torus action in such a way that there are finitely many irreducible
invariant subvarieties. If it is smooth we also define the equivariant Euler
series and proved it is rational, we relate both series and compute some
classical examples. The projective space {\bf P}^n, the blow up of {\bf P}^n at
a point, Hirzebruch surfaces, the product of {\bf P}^n with {\bf P}^m.


title: Some remarks on the Kronheimer-Mrowka classes of algebraic surfaces

abstract:
  Define the Donaldson series of a simply connected 4-manifold by
  q(X) = \sum_d q_d(X)/d!
  Recently Kronheimer and Mroka have announced the result that the Donaldson
series of so called simple 4-manifolds can be written as
  q(X) = e^{Q/2}\sum_{i=1}^p a_i e^{K_i}
  where $Q$ is the intersection form and the $K_i \in H^2(X,\Z)$ are the {\it
Kronheimer-Mrowka classes}. We prove that for simple simply connected algebraic
surfaces the $K_i$ are algebraic classes and that they are closely related to
the canonical class $K_X$. For simple simply connected minimal surfaces of
general type we prove $K_i^2 \le K_X^2$ with equality if and only if $K_i = \pm
K_X$.
  Remark: although no gauge theory is used in this paper it should have a cross
reference with the as yet non existent e-print service for low dimensional
topology.


title:
Vector bundles and $SO(3)$ invariants for elliptic surfaces III: The
  case of odd fiber degree

abstract:
  This paper, the last in a series of three, studies vector bundles on an
elliptic surface whose determinant has odd intersection number with a general
fiber and uses this study to calculate certain coefficients of Donaldson
polynomials.


title: The existence of higher logarithms

abstract:
  In this paper we prove the existence of all higher logarithms as multivalued
and ordinary Deligne cohomology classes.


title: Unobstructedness of Calabi-Yau OrbiKleinfolds

abstract:
  We show that Calabi-Yau spaces with certain types of hypersurface- quotient
singularities have unobstructed deformations. This applies in particular to all
Calabi-Yau orbifolds nonsingular in codimension 2.


title: Theoremes de connexites et varietes abeliennes

abstract:
  We prove a connexity theorem for abelian varieties in characteristic $0$: if
$X$ is an abelian variety and $V\rightarrow X$ and $W\rightarrow X$ two
morphisms, then, under certain hypotheses, the fiber product of $V$ and $W$
over $X$ is connected. This theorem has several consequences: the algebraic
fundamental groups of a simple abelian variety $X$ and of a normal subvariety
$V$ of $X$ of dimension $> dim(X)/2$ are isomorphic. The same holds when $V$ is
a finite cover of degree $\le dim(X)$ of $X$. (The only difference between this
replacement and the original submission is that one of the conjectures is now
proved).


title: A family of \'etale coverings of the affine line

abstract:
  In the note we construct a family of \'etale coverings of the affine line.
More specifically, let $F$ be a finite field of characteristic $p$ and suppose
that the cardinality of $F$ is at least 4. Let $A = F[T]$ be the polynomial
ring in one variable $T$, $K=F(T)$. Let $K_\infty$ be the completion of $K$
along the valuation given by $1/T$, and let $C$ be the completion of the
algebraic closure of $K_\infty$. We prove in this note that there is a
continous surjection
  $$\pi_1^{alg}(\A^1_C) \to \lim_{\leftarrow \atop I} SL_2( A/I )/{\pm 1},$$
where $\pi_1^{alg}(\A^1_C)$ is the algebraic fundamental group of the affine
line over $C$, and the inverse limit on the right (above) is taken over all
nonzero proper ideals in $A=F[T]$.
  We use the theory of Drinfel'd modular curves to obtain these coverings.


title: Conformal blocks and generalized theta functions

abstract:
  Let M(r) be the moduli space of rank r vector bundles with trivial
determinant on a Riemann surface X . This space carries a natural line bundle,
the determinant line bundle L . We describe a canonical isomorphism of the
space of global sections of L^k with a space known in conformal field theory as
the ``space of conformal blocks", which is defined in terms of representations
of the Lie algebra sl(r, C((z))).


title: Generic Singularities of Holomorphic Foliations

abstract:
  A holomorphic foliation is defined as an integrable coherent subsheaf of the
tangent sheaf. The structure of the leaves around a singularity is read off
from the structure of the stalks. This was done by Baum when the dimension of
the singular set is one less than the leaf dimension. In this article we study
the general case and construct some examples.


title:
Differential-geometric methods for the lifting problem and linear
  systems on plane curves

abstract:
  Let $X$ be an integral projective variety of codimension two, degree $d$ and
dimension $r$ and $Y$ be its general hyperplane section. The problem of lifting
generators of minimal degree $\sigma$ from the homogeneous ideal of $Y$ to the
homogeneous ideal of $X$ is studied. A conjecture is given in terms of $d$, $r$
and $\sigma$; it is proved in the cases $r=1,2,3$. A description is given of
linear systems on smooth plane curves whose dimension is almost maximal.


title: Conic bundles in projective fourspace

abstract:
  P. Ellia and G.Sacchiero have shown that if $S$ is a smooth surface in $\Pn
4$ which is ruled in conics, then $S$ has degree 4 or 5. In this paper we give
a proof of this result combining the ideas of Ellia and Sacchiero as they are
used in the paper of the second author on plane curve fibrations and the recent
work of G. Fl\o ystad and the first author bounding the degree of smooth
surfaces in $\Pn 4$ not of general type.


title: The Monomial-Divisor Mirror Map

abstract:
  For each family of Calabi-Yau hypersurfaces in toric varieties, Batyrev has
proposed a possible mirror partner (which is also a family of Calabi-Yau
hypersurfaces). We explain a natural construction of the isomorphism between
certain Hodge groups of these hypersurfaces, as predicted by mirror symmetry,
which we call the monomial-divisor mirror map. We indicate how this map can be
interpreted as the differential of the expected mirror isomorphism between the
moduli spaces of the two Calabi-Yau manifolds. We formulate a very precise
conjecture about the form of that mirror isomorphism, which when combined with
some earlier conjectures of the third author would completely specify it. We
then conclude that the moduli spaces of the nonlinear sigma models whose
targets are the different birational models of a Calabi-Yau space should be
connected by analytic continuation, and that further analytic continuation
should lead to moduli spaces of other kinds of conformal field theories. (This
last conclusion was first drawn by Witten.)


title:
Towards the Mirror Symmetry for Calabi-Yau Complete intersections in
  Gorenstein Toric Fano Varieties

abstract:
  We propose a combinatorical duality for lattice polyhedra which conjecturally
gives rise to the pairs of mirror symmetric families of Calabi-Yau complete
intersections in toric Fano varieties with Gorenstein singularities. Our
construction is a generalization of the polar duality proposed by Batyrev for
the case of hypersurfaces.


title: General hyperplane sections of nonsingular flops in dimension 3

abstract: We give a simple proof of a theorem of Katz and Morrison.


title:
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric
  Varieties

abstract:
  We consider families ${\cal F}(\Delta)$ consisting of complex
$(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular
affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed
$n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus
${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton
polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then
the dual, or polar, polyhedron $\Delta^*$ in the dual space defines another
family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the
remarkable duality between two {\em different families} of Calabi-Yau
varieties. It is shown that the properties of this duality coincide with the
properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau
$3$-folds. Our method allows to construct many new examples of Calabi-Yau
$3$-folds and new candidats for their mirrors which were previously unknown for
physicists. We conjecture that there exists an isomorphism between two
conformal field theories corresponding to Calabi-Yau varieties from two
families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.


title: Quantum Cohomology Rings of Toric Manifolds

abstract:
  We compute the quantum cohomology ring $H^*_{\varphi}({\bf P}, {\bf C})$ of
an arbitrary $d$-dimensional smooth projective toric manifold ${\bf
P}_{\Sigma}$ associated with a fan $\Sigma$. The multiplicative structure of
$H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ depends on the choice of an element
$avarphi$ in the ordinary cohomology group $H^2({\bf P}_{\Sigma}, {\bf C})$.
There are many properties of the quantum cohomology rings $H^*_{\varphi}({\bf
P}_{\Sigma}, {\bf C})$ which are supposed to be valid for quantum cohomology
rings of K\"ahler manifolds


title:
Teichm\"uller Theory and the Universal Period Mapping via Quantum
  Calculus and the $H^{1/2}$ Space on the Circle

abstract:
  The Universal Teichm\"uller Space, $T(1)$, is a universal parameter space for
all Riemann surfaces. In earlier work of the first author it was shown that one
can canonically associate infinite- dimensional period matrices to the
coadjoint orbit manifold $Diff(S^1)/Mobius(S^1)$ -- which resides within $T(1)$
as the (Kirillov-Kostant) submanifold of ``smooth points'' of $T(1)$. We now
extend that period mapping $\Pi$ to the entire Universal Teichm\"uller space
utilizing the theory of the Sobolev space $H^{1/2}(S^1)$. $\Pi$ is an
equivariant injective holomorphic immersion of $T(1)$ into Universal Siegel
Space, and we describe the Schottky locus utilizing Connes' ``quantum
calculus''. There are connections to string theory. Universal Teichm\"uller
Space contains also the separable complex submanifold $T(H_\infty)$ -- the
Teichm\"uller space of the universal hyperbolic lamination. Genus-independent
constructions like the universal period mapping proceed naturally to live on
this completion of the classical Teichm\"uller spaces. We show that
$T(H_\infty)$ carries a natural convergent Weil-Petersson pairing.

categories: alg-geom funct-an hep-th math.AG math.FA


title: Torsion Sections of Elliptic Surfaces

abstract:
  Given a torsion section of a semistable elliptic surface, we prove
equidistribution results for the components of singular fibers which are hit by
the section, and for the root of unity (identifying the zero component with
${\Bbb C}$) which is hit by the section in case the section hits the zero
component.


title: Initial ideals, Veronese subrings, and rates of algebras

abstract:
  We show that high Veronese subrings of any commutative graded ring have a
Grobner basis with all relations of degree 2. (The d-th Veronese subring of a
ring A_0 + A_1 + A_2 + ... is the ring A_0 + A_d + A_{2d} + ...; ``high'' means
we take d sufficiently large, say at least half the regularity of the ideal
defining the original ring.) This gives another proof of Backelin's theorem
that such Veronese subrings are Koszul algebras (= wonderful rings), i.e., that
the minimal resolution of the residue field of such a ring is linear.


title: A Propos de l'Existence de Fibr\'es Stables sur les Surfaces

abstract:
  We prove the existence (in characteristic 0) on every polarized (smooth,
projective and connected) surface of stable bundles of rank $r\geq 2$,
arbitrary first Chern class and large enough $c_2$.


title: On Banica sheaves and Fano manifolds

abstract:
  In the present paper we discuss coherent sheaves of rank > 1 whose
projectivization gives rise to smooth varieties - varieties of this type are
also called smooth scrolls. We prove some basic properties of these varieties
and we give some pertinent examples. We classify Fano manifolds of large index
which are of this type.


title:
Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano
  Threefolds

abstract:
  In this article we exhibit certain projective degenerations of smooth $K3$
surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by the
hyperplane class), to a union of two rational normal scrolls, and also to a
union of planes. As a consequence we prove that the general hyperplane section
of such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$.
We also prove that the general such hyperplane section lies on a unique $K3$
surface, up to projectivities. Finally we present a new approach to the
classification of prime Fano threefolds of index one, which does not rely on
the existence of a line.


title: Geometric height inequality on varieties with ample cotangent bundles

abstract:
  Let F be a function field of one variable over an algebraically closed field
of characteristic zero, X a geometrically irreducible smooth projective variety
over F, and L a line bundle on X. In this note, we will prove that if the
contangent bundle of X is ample and X is non-isotrivial, then there are a
proper closed algebraic set Y of X and a constant A > 0 such that h_L(P) <= A
d(P) + O(1) for all P \in X(\bar{F}) \ Y(\bar{F}), where h_L(P) is a geometric
height of P with respect to L and d(P) is the geometric logarithmic
discriminant of P. As corollary of the above height inequality, we can recover
Noguchi's theorem, i.e. there is a non-empty Zariski open set U of X with U(F)
= \emptyset.


title: On discrete Zariski-dense subgroups of algebraic groups

abstract:
  We investigate for which linear-algebraic groups (over the complex numbers or
any local field) there exists subgroups which are dense in the Zariski
topology, but discrete in the Hausdorff topology. For instance, such subgroups
exist for every non-solvable complex group.


title:
Irreducibility of Moduli Spaces of Vector Bundles on Birationally Ruled
  Surfaces

abstract:
  Let $S$ be a birationally ruled surface. We show that the moduli schemes
$M_S(r,c_1,c_2)$ of semistable sheaves on $S$ of rank $r$ and Chern classes
$c_1$ and $c_2$ are irreducible for all $(r,c_1,c_2)$ provided the polarization
of $S$ used satisfies a simple numerical condition. This is accomplished by
proving that the stacks of prioritary sheaves on $S$ of fixed rank and Chern
classes are smooth and irreducible.


title:
Surfaces of Degree 10 in the Projective Fourspace via Linear Systems and
  Linkage

abstract:
  The paper discusses the classification of surfaces of degree 10 and sectional
genus 9 and 10. The surfaces of degree at most 9 are described through
classical work dating from the last century up to recent years, while surfaces
of degree 10 and other sectional genera are studied elsewhere.
  We use relations between multisecants, linear systems, syzygies and linkage
to describe the geometry of each surface. We want in fact to stress the
importance of multisecants and syzygies for the study of these surfaces.
Adjunction, which provided efficient arguments for the classification of
surfaces of smaller degrees, here appears to be less effective and will play
almost no role in the proofs.
  We show that there are 8 different families of smooth surfaces of degree 10
and sectional genus 9 and 10. The families are determined by numerical data
such as the sectional genus, the Euler characteristic, the number of 6-secants
to the surface and the number of 5-secants to the surface which meet a general
plane. For each type we describe the linear system giving the embedding in P^4,
the resolution of the ideal, the geometry of the surface in terms of curves on
the surface and hypersurfaces containing the surface, and the liaison class; in
particular we find minimal elements in the even liaison class. Each type
corresponds to an irreducible, unirational component of the Hilbert scheme, and
the dimension is computed.


title: On the Picard Number of Fano 3-Folds with Terminal Singularities

abstract:
  We prove the following main result: Let X be a Fano 3-fold with terminal
Q-factorial singularities and X does not have a small extremal ray and a face
of Kodaira dimension 1 or 2 for Mori polyhedron of X. Then the Picard number
\rho (X) < 8.


title: Spin polynomial invariants for Dolgachev surfaces

abstract:
  We consider the spin polynomial invariants for bundles with c_2=2 and c_1 =
K_S + 2nk a rational mutiple of the canonical divisor on a Dolgacev surface. It
is shown that the chamber structure can be controlled so that the polynomials
give diffeomorphism invariants of Dolgachev surfaces of the form q_S(n) =
a(n)Q^2 + b(n)Qk^2 + c(n)k^4. The two leading coefficients are computed.


title: On Brauer Groups of Real Enriques Surfaces

abstract:
  Let Y be a real Enriques surface, _2Br(Y) the subgroup of elements of order 2
of Br(Y), and s, s_{or}, and s_{nor} the number of all connected, connected
orientable, and connected non-orientable components of Y(R) respectively. Using
universal covering K3-surface X of Y, we connect dim _2Br(Y) with the s, s_{or}
and s_{nor}. As a geometric corollary of our considerations, we show that s \le
6 and s_{nor} \le 4.


title: On the Brauer Group of Real Algebraic Surfaces

abstract:
  Let X be a real projective algebraic manifold, s numerates connected
components of X(R) and _2Br(X) the subgroup of elements of order 2 of the
cohomological Brauer group Br(X). We study the natural homomorphism \xi :
_2Br(X) \to (Z/2)^s and prove that \xi is epimorphic if H^3(X(C)/G;Z/2) \to
H^3(X(R);Z/2) is injective. Here G=Gal(C/R). For an algebraic surface X with
H^3(X(C)/G;Z/2)=0 and X(R)\not=\emptyset, we give a formula for dim _2Br(X). As
a corollary, for a real Enriques surfaces Y, the \xi is epimorphic and dim
_2Br(Y)=2s-1 if both liftings of the antiholomorphic involution of Y to the
universal covering K3- surface X have non-empty sets of real points (this is
the general case). For this case, we also give a formula for the number s_{nor}
of non-orientable components of Y which is very important for the topological
classification of real Enriques surfaces.


title: On the Topological Classification of Real Enriques Surfaces. I

abstract:
  This note contains preliminary calculation of topological types or real
Enriques surfaces. We realize 59 topological types of real Enriques surfaces
(Theorem 6) and show that all other topological types belong to the list of 21
topological types (Theorem 7). In fact, our calculation contains much more
information which is probably useful to constract or prohibit unknown
topological types.


title:
Remarks on Ein-Lazarsfeld criterion of spannedness of adjoint bundles of
  polarized threefolds

abstract:
  Let B be a nef and big line bundle on a smooth complex threefold X with
canonical bundle K. Let x be a point on X and suppose that BC\ge3 for any curve
C passing x, B^2S\ge7 for any surface S containing x, and B^3\ge51. Then K+B is
spanned at x. (Ein-Lazarsfeld proved the assertion assuming B^3\ge92.)
Corollary: K+3L is spanned if L is an ample line bundle with L^3>1.


title: The Functor of a Smooth Toric Variety

abstract:
  A map Y -> P^n is determined by a line bundle quotient of (O_Y)^{n+1}. In
this paper, we generalize this description to the case of maps from Y to an
arbitrary smooth toric variety. The data needed to determine such a map
consists of a collection of line bundles on Y together with a section of each
line bundle. Further, the line bundles must satisfy certain compatbility
conditions, and the sections must be nondegenerate in an appropriate sense. In
the case of maps from P^m to a smooth toric variety, we get an especially
simple description that generalizes the usual way of specifying maps between
projective spaces in terms of homogeneous polynomials (of the same degree) that
don't vanish simultaneously.


title:
Complete ideals defined by sign conditions and the real spectrum of a
  two-dimensional local ring

abstract:
  This paper is about the local geometry of a real surfaces. It introduces
machinery for studying families of subsets which are determined by conditions
which are similar to base conditions, but also involve
positivity/non-negativity. The methods used are the real spectrum and Zariski's
theory of complete ideals.


title: Thickening Calabi-Yau moduli spaces

abstract:
  We describe a kind of deformation of the anti-DeRham algebra on a Calabi-Yau
manifold $X$. These are in 1-1 correspondence with the total cohomology $\oplus
H^i (X, \C)$.


title: On Koszul property of the homogeneous coordinate ring of a curve

abstract:
  The following corollary has been added: for general tetragonal curve $C$ of
genus $g\ge 9$ the homogeneous coordinate ring of $C$ defined by the line
bundle $K(-T)$, where $K$ is the canonical class, $T$ is the tetragonal series,
is Koszul. Also some misprints are corrected.


title: On the Hilbert Schemes of Canonically-Embedded Curves of Genus 5 and 6

abstract:
  A canonically-embedded curve of genus $g$ is a pure 1-dimensional,
non-degenerate subscheme $C$ of ${\bf P}^{g-1}$ over an algebraically closed
field $k$, for which ${\cal O}_C(1) \cong \omega_C$, (the dualizing sheaf)$ and
$h^0(C, {\cal O}_C) = 1$, $h^0(C, \omega_C) = g$. The singularities of $C$ (if
any) are Gorenstein, and $C$ is connected of degree $2g-2$ and arithmetic genus
$g$. In a recent paper, Schreyer has proved that Petri's normalization of the
homogeneous ideal $I(C)$ of a smooth canonically-embedded curve can be also
carried out for singular curves, provided that the curve has a simple
$(g-2)$-secant (a linear ${\bf P}^{g-3}$ intersecting $C$ transversely at
exactly $g-2$ (smooth) points). We use the Petri normalization to study the
Hilbert scheme of curves of degree $2g-2$ and arithmetic genus $g$ in ${\bf
P}^{g-1}$ in the low-genus cases $g = 5,6$. The main results are that the
Hilbert points of all curves for which Petri's approach applies lie on one
irreducible component of the Hilbert scheme in these cases. (The same is not
true for $g$ sufficiently large.)


title: An algorithmic criterion for basicness in dimension 2

abstract:
  We give a constructive procedure to check basicness of open (or closed)
semialgebraic sets in a compact, non singular, real algebraic surface $X$. It
is rather clear that if a semialgebraic set $S$ can be separated from each
connected component of $X\setminus(S\cup\frz S)$ (when $\frz S$ stands for the
Zariski closure of $(\ol S\setminus{\rm Int}(S))\cap{\rm Reg}(X)$), then $S$ is
basic. This leads to associate to $S$ a finite family of sign distributions on
$X\setminus\frz S$; we prove the equivalence between basicness and two
properties of these distributions, which can be tested by an algorithm. There
is a close relation between these two properties and the behaviour of fans in
the algebraic functions field of $X$ associated to a real prime divisor, which
gives an easy proof, for a general surface $X$, of the well known 4-elements
fan's criterion for basicness (Brocker, Andradas-Ruiz). Furthermore, if the
criterion fails, using the description of fans in dimension 2, we find an
algorithmic method to exhibit the failure. Finally, exploiting this thecnics of
sign distribution we give one improvement of the 4-elements fan's criterion of
Brocker to check if a semialgebraic set is principal.


title: Hyperbolicity of the complement of plane algebraic curves

abstract:
  The paper is a contribution of the conjecture of Kobayashi that the
complement of a generic plain curve of degree at least five is hyperbolic. The
main result is that the complement of a generic configuration of three quadrics
is hyperbolic and hyperbolically embedded as well as the complement of two
quadrics and a line.


title:
On the Hyperbolicity of the Complements of Curves in Algebraic Surfaces:
  The Three Component Case

abstract:
  The paper is a contribution to the conjecture of Kobayashi that the
complement of a generic curve in the projective plane is hyperbolic, provided
the degree is at least five. Previously the authors treated the cases of two
quadrics and a line and three quadrics. The main results are Let C be the union
of three curves in P_2 whose degrees are at least two, one of which is at least
three. Then for generic such configurations the complement of C is hyperbolic
and hyperbolically embedded. The same statement holds for complements of curves
in generic hypersurfaces X of degree at least five and curves which are
intersections of X with hypersurfaces of degree at least five. Furthermore
results are shown for curves on surfaces with picard number one.


title: Rational curves on Calabi-Yau threefolds

abstract:
  This note is a survey of the enumerative geometry of rational curves on
Calabi-Yau threefolds, based on a talk given by the author at the May 1991
Workshop on Mirror Symmetry at MSRI. An earlier version appeared in "Essays on
Mirror Manifolds"; this version corrects typographical errors that appeared in
print, gives a brief update of related progress during the last two years in
the form of footnotes, and has more and updated references. (To appear in the
second edition of Essays on Mirror Manifolds.)


title: On the Harder-Narasimhan Filtration for Coherent Sheaves on P2: I

abstract:
  Let E be a torsion-free sheaf on P2. We give an effective method which uses
the Hilbert function of E to construct a weak version of the Harder-Narasimhan
filtration of a torsion-free sheaf on P2 subject only to the condition that E
be sufficiently general among sheaves with that Hilbert function. This
algorithm uses on a generalization of Davis' decomposition lemma to higher
rank.


title:
Components of the Stack of Torsion-Free Sheaves of Rank 2 on Ruled
  Surfaces

abstract:
  Let S be a ruled surface without sections of negative self-intersection. We
classify the irreducible components of the moduli stack of torsion-free sheaves
of rank 2 sheaves on S. We also classify the irreducible components of the
Brill-Noether loci in Hilb^N(P1xP1) given by W_N^0(D)={[X] | h^1(I_X(D)) >= 1 }
for D an effective divisor class. Our methods are also applicable to P2 giving
new proofs of theorems of Stromme (slightly extended) and Coppo.


title: Enumeration of $n$-fold tangent hyperplanes to a surface

abstract:
  For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal
curves in an $n-$dimensional linear system on a smooth, projective surface.
This yields in particular the numbers of rational curves in the system of
hyperplane sections of a generic $K3-$surface imbedded in \p{n} by a complete
system of curves of genus $n$ as well as the number {\bf17,601,000} of rational
({\em singular}) plane quintic curves in a generic quintic threefold.


title: Binomial Ideals

abstract:
  We investigate the structure of ideals generated by binomials (polynomials
with at most two terms) and the schemes and varieties associated to them. The
class of binomial ideals contains many classical examples from algebraic
geometry, and it has numerous applications within and beyond pure mathematics.
The ideals defining toric varieties are precisely the binomial prime ideals.
Our main results concern primary decomposition: If $I$ is a binomial ideal then
the radical, associated primes, and isolated primary components of $I$ are
again binomial, and $I$ admits primary decompositions in terms of binomial
primary ideals. A geometric characterization is given for the affine algebraic
sets that can be defined by binomials. Our structural results yield
sparsity-preserving algorithms for finding the radical and primary
decomposition of a binomial ideal.


title:
Holomorphic functions on an algebraic group invariant under a
  Zariski-dense subgroup

abstract:
  Let G be complex linear-algebraic group, H a subgroup, which is dense in G in
the Zariski-topology. Assume that G/[G,G] is reductive and furthermore that (1)
G is solvable, or (2) the semisimple elements in G'=[G,G] are dense. Then every
H-invariant holomorphic function on G is constant. If G=G', furthermore every
H-invariant meromorphic or plurisubharmonic function is constant. Finally an
example of Margulis is used to show the existence of an algebraic group G with
G=G' such that there exists a Zariski-dense discrete subgroup without any
semisimple element.


title: The Fundamental Group of Some Siegel Modular Threefolds

abstract:
  Let A be the moduli space of (1,p)-polarised abelian surfaces with a level
structure, for p an odd prime. Let X be a desingularisation of any algebraic
compactification of A. Then X is simply-connected.


title: Matrix identities connected with the jacobian conjecture

abstract:
  We give necessary and sufficient conditions, in the form of matrix
identities, for a polynomial f in C[X,Y] to be a component of a polynomial
automorphism of C^2 and to be a component of a Keller polynomial mapping of
C^2, respectively ((f,g) is a Keller polynomial mapping iff Jac(f,g) = 1).


title:
Stable cohomology of the mapping class group with symplectic
  coefficients and of the universal Abel-Jacobi map

abstract:
  This replacement corrects statement and proof of the main result. Also, a
section on the universal Abel-Jacobi map has been added.


title:
A common generalization of the Fr\"olicher-Nijenhuis bracket and the
  Schouten bracket for symmetry multi vector fields

abstract:
  There is a canonical mapping from the space of sections of the bundle $\wedge
T^\ast M\otimes ST\ M$ to $\Omega(T^\ast M ; T(T^\ast M))$. It is shown that
this is a homomorphism on $\Omega(M;TM) for the Fr\"olicher-Nijenhuis brackets,
and also on $\Gamma(ST\ M)$ for the Schouten bracket of symmetric multi vector
fields. But the whole image is not a subalgebra for the Fr\"olicher-Nijenhuis
bracket on $\Omega(T^\ast M;T(T^\ast M))$.


title: A blow-up construction and graph coloring

abstract:
  Given a graph G (or more generally a matroid embedded in a projective space),
we construct a sequence of varieties whose geometry encodes combinatorial
information about G. For example, the chromatic polynomial of G (giving at each
m>0 the number of colorings of G with m colors, such that no adjacent vertices
are assigned the same color) can be computed as an intersection product between
certain classes on these varieties, and other information such as Crapo's
invariant find a very natural geometric counterpart. The note presents this
construction, and gives `geometric' proofs of a number of standard
combinatorial results on the chromatic polynomial.


title: Stable triples, equivariant bundles and dimensional reduction

abstract:
  This is a resubmission of preprint 9401008 , which has some TeXnical errors
introduced by the "reform" procedure (designed to avoid precisely these
problems!). The original can be formatted by editing out the messages "%%
following line cannot be broken..." at line 1674. This problem has been
corrected in the current version. The content of the paper has not been changed
in any way. My aplogies for any inconvenience caused. SB


title: The Connectedness of Space Curve Invariants

abstract:
  It is a result of Gruson and Peskine that the invariants of a set points in
$\ptwo$ in general position are connected. Associated to a space curve there
are sequences of invariants which generalize the invariants of points in
$\ptwo$. The main result of this paper is to show that the invariants of
reduced, irreducible, non-degenerate curves in $\pthree$ also satisfy a
connectedness property. This result greatly restricts the kinds of Borel-fixed
monomial ideals which can occur as generic initial ideals of such curves and
thus gives us more control over their Hilbert functions.


title:
Diagram Method for 3-Folds and its Application to Kahler Cone and Picard
  Number of Calabi-Yau 3-Folds, I. with Appendix by Vyacheslav V. Shokurov:
  "Anticanonical boundedness for curves"

abstract:
  We prove the general diagram method theorem valid for the quite large class
of 3-folds with Q-factorial singularities (see Basic Theorem 1.3.2 and also
Theorem 2.2.6). This gives the generalization of our results about Fano 3-folds
with Q-factorial terminal singularities (Preprint alg-geom/9311007).
  As an application, we get the following result about 3-dimensional Calabi-Yau
manifolds X: Assume that the Picard number \rho (X) > 93. Then one of two cases
(i) or (ii) holds:
  (i) there exists a small extremal ray on X;
  (ii) There exists a nef element h such that h^3=0 (thus, the nef cone NEF(X)
and the cubic intersection hypersurface W_X have a common point; here, we don't
claim that h is rational!).
  As a corollary, we get: Let X be a 3-dimensional Calabi-Yau manifold. Assume
that the nef cone NEF(X) is finite polyhedral and X does not have a small
extremal ray. Then there exists a rational nef element h with h^3=0 if
\rho(X)>93.
  To prove these results about Calabi-Yau manifolds, we also use one result of
V.V. Shokurov on the length of divisorial extremal rays (see Appendix by V.V.
Shokurov). Thus, one should consider the results on Calabi-Yau 3-folds above as
our common results with V.V. Shokurov.
  We also discuss generalization of results above to so called Q-factorial
models of Calabi-Yau 3-folds, which sometimes permits to involve non-polyhedral
case and small extremal rays to the game.


title:
The theta divisior of the bidegree (2,2) threefold in ${\bf P}^{2}
  \times {\bf P}^{2}$

abstract:
  Let T be a general bidegree (2,2) divisor in the product of two projective
planes. Recently A.Verra proved that the existence of two conic bundle
structures (c.b.s.) on T implies a new counterexample to the Torelli theorem
for Prym varieties. Let J(T) be the jacobian of T. In this paper we prove that
any of the two c.b.s. on T admits a parametrisation of the theta divisor of
J(T) by the Abel-Jacobi image of a special family of elliptic curves of degree
9 (minimal sections of the given c.b.s.) on T. This result is an analogue of
the well-known Riemann theorem for curves. In particular, this implies some
results about K3 surfaces and plane sextics with vanishing theta-null. Further
we use once again the geometry of curves on T, in order to prove the Torelli
theorem for the bidegree (2,2) threefolds.


title: Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds

abstract:
  We introduce a special class of convex rational polyhedral cones which allows
to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$,
where $r$ is a positive integer and d is the dimension of critical string vacua
with central chatge $c = 3d$. It is conjectured that the natural combinatorial
duality satisfies by these cones corresponds to the mirror involution. Using
the theory of toric varieties, we show that our conjecture includes as special
cases all already known examples of mirror pairs proposed by physicists and
agrees with previous conjectures of the authors concerning explicit
constructions of mirror manifolds. In particular we obtain a mathematical
framework which explains the construction of mirrors of rigid Calabi-Yau
manifolds.


title: The combinatorics of the Verlinde formulas

abstract:
  A residue formula is given for the Verlinde formula, which allows one to
calculate its coefficients as a polynomial in the level and connects it to the
Riemann-Roch formula on the moduli space of vector bundles on a curve.


title: Boundedness theorem for Fano log-threefolds

abstract:
  The main purpose of this article is to prove that the family of all Fano
threefolds with log-terminal singularities with bounded index is bounded.


title: The Tangent Space at a Special Symplectic Instanton Bundle on $P_{2n+1}$

abstract:
  Mathematical instanton bundles on $ P_3$ have their analogues in rank--$2n$
instanton bundles on odd dimensional projective spaces $ P_{2n+1}$. The
families of special instanton bundles on these spaces generalize the special
'tHooft bundles on $ P_3$. We prove that for a special symplectic instanton
bundle $ E$ on $ P_{2n+1}$ with $c_2=k$ $h^1End( E) = 4(3n-1) k +
(2n-5)(2n-1)$. Therefore the dimension of the moduli space of instanton bundles
grows linearly in $k$. The main difference with the well known case of $ P_3$
is that $h^2End( E)$ is nonzero, in fact we prove that it grows quadratically
in $k$. Special symplectic instanton bundles turn out to be singular points of
the moduli space. Such bundles $ E$ are $SL(2)$--invariant and the result is
obtained regarding the cohomology groups of $ E$ as $SL(2)$--representations.


title: On surfaces in P^4 and 3-folds in P^5

abstract:
  This is a survey on the classification of smooth surfaces in P^4 and smooth
3-folds in P^5. We recall the corresponding results arising from adjunction
theory and explain how to construct examples via syzygies. We discuss some
examples in detail and list all families of smooth non-general type surfaces in
P^4 and 3-folds in P^5 known to us.


title: Boundedness and $K^2$ for log surfaces

abstract:
  Let $\epsilon, C$ be two positive real numbers, and $\mathcal C \subset
\mathbb R$ be a DCC (descending chain condition) set. Let $(X, B = \sum b_j
B_j)$ denote a projective surface with an $\mathbb R$-divisor. Then
  (1) The class $\{X\}$ of surfaces for which there exists a divisor $B$ such
that $(X,B)$ is $\epsilon$-log terminal and $-(K_X + B)$ is nef (excluding only
those for which at the same time $K_X\equiv 0$, $B=0$, and $X$ has at worst Du
Val singularities), is bounded.
  (2) The set $\{(K_X + B)^2\}$ of squares for the semi log canonical pairs
$(X, B)$ with ample $K_X + B$ and $b_j \in \mathcal C$, is a DCC set.
  (3) The class $\{(X,B)\}$ of pairs such that $(X, B)$ is semi log canonical,
$K_X + B$ is ample, $(K_X + B)^2 = C$ and $b_j \in \mathcal C$, is bounded.


title: Variation of Geometric Invariant Theory Quotients

abstract:
  Geometric Invariant Theory gives a method for constructing quotients for
group actions on algebraic varieties which in many cases appear as moduli
spaces parametrizing isomorphism classes of geometric objects (vector bundles,
polarized varieties, etc.). The quotient depends on a choice of an ample
linearized line bundle. Two choices are equivalent if they give rise to
identical quotients. A priori, there are infinitely many choices since there
are infinitely many isomorphism classes of linearized ample line bundles. Hence
several fundamental questions naturally arise. Is the set of equivalence
classes, and hence the set of non-isomorphic quotients, finite? How does the
quotient vary under change of the equivalence class? In this paper we give
partial answers to these questions in the case of actions of reductive
algebraic groups on nonsingular projective algebraic varieties. We shall show
that among ample line bundles which give projective geometric quotients there
are only finitely many equivalence classes. These classes span certain convex
subsets (chambers) in a certain convex cone in Euclidean space; and when we
cross a wall separating one chamber from another, the corresponding quotient
undergoes a birational transformation which is similar to a Mori flip.


title: On the Locus of Hodge Classes

abstract:
  Let $f: X \rightarrow S$ be a family of non singular projective varieties
parametrized by a complex algebraic variety $S$. Fix $s \in S$, an integer $p$,
and a class $h \in {\rm H}^{2p}(X_s,\Z)$ of Hodge type $(p,p)$. We show that
the locus, on $S$, where $h$ remains of type $(p,p)$ is algebraic. This result,
which in the geometric case would follow from the rational Hodge conjecture, is
obtained in the setting of variations of Hodge structures.


title: On Polarized Manifolds of Sectional Genus Three

abstract:
  This paper is an attempt for the classification of polarized manifolds of
sectional genus $g=3$ and dimension $n\geq 3$. As in the case of $g\leq
2$,which was classified by T.Fujita,we use Mori-Kawamata theory. The
classification result of $g=3$ and $n=2$, which was obtained by H.Maeda,is also
used essentially. Although our results are far from being complete, they are
very similar to those in case $g=2$.


title:
Degree of the generalized Pl\"ucker embedding of a Quot scheme and
  Quantum cohomology

abstract:
  We compute the degree of the generalized Pl\"ucker embedding $\kappa$ of a
Quot scheme $X$ over $\PP^1$. The space $X$ can also be considered as a
compactification of the space of algebraic maps of a fixed degree from $\PP^1$
to the Grassmanian $\rm{Grass}(m,n)$. Then the degree of the embedded variety
$\kappa (X)$ can be interpreted as an intersection product of pullbacks of
cohomology classes from $\rm{Grass}(m,n)$ through the map $\psi$ that evaluates
a map from $\PP^1$ at a point $x\in \PP^1$. We show that our formula for the
degree verifies the formula for these intersection products predicted by
physicists through Quantum cohomology~\cite{va92}~\cite{in91}~\cite{wi94}. We
arrive at the degree by proving a version of the classical Pieri's formula on
the variety $X$, using a cell decomposition of a space that lies in between $X$
and $\kappa (X)$.


title:
Factorization theorems for the representations of the fundamental groups
  of quasiprojective varieties and some applications

abstract:
  In this paper, using Gromov-Jost-Korevaar-Schoen technique of harmonic maps
to nonpositively curved targets, we study the representations of the
fundamental groups of quasiprojective varieties. As an application of the above
considerations we give a proof of a weak version of the Shafarevich Conjecture.


title: Faltings modular height and self-intersection of dualizing sheaf

abstract:
  Let K be a number field, O_K the ring of integers of K and X a stable curve
over O_K of genus g >= 2. In this note, we will prove a strict inequality
  ( (K_{X/S})^2 / [K : Q] ) > Height_{Fal}(J(X_K)),
  where $K_{X/S}$ is the canonically metrized dualizing sheaf of X over S =
Spec(O_K) and Height_{Fal}(J(X_K)) is the Faltings modular height of the
Jacobian of X_K. As corollary, for any constant A, the set of all stable curves
X over O_K with
  ( (K_{X/S})^2 / [K : Q] ) <= A
  is finite under the following equivalence. For stable curves X and Y, X is
equivalent to Y if X is isomorphic to Y over O_{K'} for some finite extension
field K' of K.


title:
The Deformation Space of Calabi-Yau n-folds with canonical singularities
  can be obstructed

abstract:
  This paper gives a simple example of a family of Calabi-Yaus of any dimension
with canonical singularities of dimension one, whose Kuranishi space is
singular. Thus the Bogomolov-Tian-Todorov unobstructedness theorem is not true
for Calabi-Yaus with canonical singularities.


title: Residual Intersections and Some Applications

abstract:
  We give a new residual intersection decomposition for the refined
intersection products of Fulton-MacPherson. Our formula refines the celebrated
residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The
new decomposition is more likely to be compatible with the canonical
decomposition of the intersection products and each term in the decomposition
thus has simple geometric meaning. Our study is motivated by its applications
to some geometric problems. In particular, we use the decomposition to find the
distribution of limiting linear subspaces in degenerations of hypersurfaces. A
family of identities for characteristic classes of vector bundles is also
obtained as another consequence. This paper will appear in Duke Math. Jour.


title: Intersection Theory on Toric Varieties

abstract:
  The operational Chow cohomology classes of a complete toric variety are
identified with certain functions, called Minkowski weights, on the
corresponding fan. The natural product of Chow cohomology classes makes the
Minkowski weights into a commutative ring; the product is computed by a
displacement in the lattice, which corresponds to a deformation in the toric
variety. We show that, with rational coefficients, this ring embeds in
McMullen's polytope algebra, and that the polytope algebra is the direct limit
of these Chow rings, over all compactifications of a given torus. In the
nonsingular case, the Minkowski weight corresponding to the Todd class is
related to a certain Ehrhart polynomial.


title: Toric Q-Gorenstein Singularities

abstract:
  For an affine, toric Q-Gorenstein variety Y (given by a lattice polytope Q)
the vector space T^1 of infinitesimal deformations is related to the
complexified vector spaces of rational Minkowski summands of faces of Q.
Moreover, assuming Y to be an isolated, at least 3-dimensional singularity, Y
will be rigid unless it is even Gorenstein and dim Y=3 (dim Q=2). For this
particular case, so-called toric deformations of Y correspond to Minkowski
decompositions of Q into a sum of lattice polygons. Their Kodaira-Spencer-map
can be interpreted in a very natural way. We regard the projective variety P(Y)
defined by the lattice polygon Q. Data concerning the deformation theory of Y
can be interpreted as data concerning the Picard group of P(Y). Finally, we
provide some examples (the cones over the toric Del Pezzo surrfaces). There is
one such variety yielding Spec C[e]/e^2 as the base space of the semi-universal
deformation.


title: The versal Deformation of an isolated toric Gorenstein Singularity

abstract:
  Given a lattice polytope Q in R^n, we define an affine scheme M(Q) that
reflects the possibilities of splitting Q into a Minkowski sum. On the other
hand, Q induces a toric Gorenstein singularity Y, and we construct a flat
family over M(Q) with Y as special fiber. In case Y has an isolated singularity
only, this family is versal. (This revised version contains the proof now.)


title: Siegel modular forms generated by invariants of cubic hypersurfaces

abstract:
  We give a geometric derivation of Schottky's equation in genus four for the
period matrices of Riemann surfaces among all period matrices. The equation
arises naturally from the singularity theory of the Gauss map on the theta
divisor, and thus generalizes for any genus $g \ge 4$ to a certain ideal of
Siegel modular forms vanishing on period matrices of Riemann surfaces. This
ideal is generated by modular forms associated to the invariants of cubic forms
in $g - 1$ variables which vanish on the Fermat cubic.


title: Hyperkaehler Embeddings II

abstract:
  In the first part, Hyperkaehler Embeddings and Holomorphic symplectic
Geometry I, we prove the following. Let $N$ be a closed analytic subvariety of
a generic deformation of a holomorphically symplectic compact manifold $M$.
Then the restriction of a holomorphic symplectic form is non-degenerate on $N$.
In particular, $N$ is even-dimensional. In present paper, we prove that there
exist a hyperkaehler metric on $M$, such that the embedding of $N$ to $M$ is
hyperkaehler.


title:
Towards a Schubert calculus for maps from a Riemann surface to a
  Grassmannian

abstract:
  The intuitive notion of the Gromov invariant for maps from a Riemann surface
to a Grassmannian is shown to agree with the definition in \cite{BDW}. Also, an
induction on the genus is proved, which extends the results of \cite{BDW} to a
computation of all Gromov invariants associated to G(2,k). This is shown to
agree with the conjectured formula of Vafa and Intriligator.


title: Torus embeddings and algebraic intersection complexes

abstract:
  We describe the intersection complex of any perversity of a toric variety
completely in terms of the associated fan. It is described by a finite complex
of finite dimensional graded vector spaces which we call graded exterior
modules. The intersection complexes of the middle perversity are treated in the
second part following this article.


title: Torus embeddings and algebraic intersection complexes, II

abstract:
  In the previous paper, we describe the intersection complexes of a toric
variety as a finite complex of graded exterior modules on the associated fan.
In this second part, we rewrite it explicitly by the barycentric subdivision of
the fan. We get the decomposition theorem of the intersection homologies for a
barycentric subdivision of a fan in the case of middle perversity. We get also
the diagonal theorems I and II. These theorems give a new proof of the
g-comnjecture on a simplicial polytope which was proved by R. Stanley.


title:
On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and
  Intriligator

abstract:
  We observe a general structure theorem for quantum cohomology rings, a
non-homogeneous version of the usual cohomology ring encoding information about
(almost holomorphic) rational curves. An application is the rigorous
computation of the quantum cohomology of Grassmannians. As purely algebraic
consequence we prove a beautiful formula of Vafa and Intriligator for
intersection numbers of certain compactifications of moduli spaces of maps from
a Riemann surface (any genus) to G(k,n) which recently has excited many
mathematicians. The formula generalizes to any Fano manifold whose cohomology
ring can be presented as complete intersection.


title: Hodge index theorem for arithmetic cycles of codimension one

abstract:
  In this note, we will give a partial answer for arithmetic analogues of
Grothendieck's standard conjectures due to H. Gillet and C. Soule. (Remark : I
changed the title of this note.)


title: Minimal discrepancies of toric singularities

abstract:
  This is the major revision. The main purpose of this paper is to prove that
minimal discrepancies of $n$-dimensional toric singularities can accumulate
only from above and only to minimal discrepancies of toric singularities of
dimension less than $n$. I also prove that some lower-dimensional minimal
discrepancies do appear as such limit.


title:
Complexes inattendus de droites de saut (Unexpected complex of jumping
  lines)

abstract:
  We prove here the following results: \begin{th} Let $E$ a rank 2 vector
bundle over ${\bf P}_3$, if $C$ is a reduced irreducible curve of ${\bf
P}_3^{\vee}$ such that $E_H$ is unstable for all $H\in C$ then $C$ is a line.
  \end{th} We define now the set $W(E)$ as the set of planes $H$ such that the
restricted bundle $E_H$ is unstable (that means non semi-stable).
  \begin{th} Let $E$ a rank 2 vector bundle over ${\bf P}_3$, with first chern
class $c_1=c_1(E)$, $L$ a line and an integer $n\ge 0$. The following
conditions are equivalent:
  \begin{description}
  \item[(i)] $L^{\vee}\subset W(E)$ and $H^0(E_H(-n+[-c_1/2]))\neq 0$ for a
general point $H\in L^{\vee}$. \item[(ii)] There exist $m>0$ and a section
$t\in H^0(E(m+[-c_1/2]))$ such that the zero variety of $t$ contains the
infinitesimal neighbourhood of order $(m+n-1)$ of $L$.
  \end{description}


title: On Chow Rings of Fine Moduli Spaces of Modules

abstract:
  Let $M$ be a complete nonsingular fine moduli space of modules over an
algebra $S$. A set of conditions is given for the Chow ring of $M$ to be
generated by the Chern classes of certain universal bundles occurring in a
projective resolution of the universal $S$-module on $M$. This result is then
applied to the varieties $G_T$ parametrizing homogeneous ideals of $k[x,y]$ of
Hilbert function $T$, to moduli spaces of representations of quivers, and
finally to moduli spaces of sheaves on ${\Bbb P}^2$, reinterpreting a result of
Ellingsrud and Str\o mme.


title: Torelli Groups and Geometry of Moduli Spaces of Curves

abstract:
  In this paper we give an exposition of Dennis Johnson's work on the first
homology of the Torelli groups and show how it can be applied, alone and in
concert with Saito's theory of Hodge modules, to study the geometry of moduli
spaces of curves. For example, we show that the picard groups of moduli spaces
of curves with a fixed level structure are finitely generated, classify all
"natural" normal functions defined over moduli spaces of curves with a fixed
level, and also "compute" the height paring between cycles over moduli spaces
of curves which are homologically trivial and disjoint over the generic point.
Several new sections have been added. These apply the results on normal
functions to prove generalizations of the classical Franchetta conjecture for
curves and abelian varieties. In one section, the monodromy group of nth roots
of the canonical bundle is computed.


title:
Groupe de Picard de la vari\'et\e de modules des
  th\^eta-caracteristiques des courbes planes

abstract:
  Nous calculons le groupe de Picard de l'espace des modules
$\Theta_{\planp}(d)$ des th\^eta-caract\'eristiques des courbes planes (non
forc\'ement lisses) de degr\'e $d$. Nous montrons que pour $d\ge 6$, le groupe
de Picard de la composante paire est engendr\'e par le fibr\'e d\'eterminant
${\cal{D}}$ et l'image r\'eciproque ${\cal{L}}$, sous le morphisme support
sch\'ematique, de ${\cal{O}}_{\proj}(1)$ o\`u $\proj$ est l'espace des courbes
de degr\'e $d$ sur $\planp$. Pour $1\le d \le 4$, ce groupe est engendr\'e
uniquement par ${\cal{L}}$, le fibr\'e d\'eterminant \'etant trivial dans ces
cas. Dans le groupe $\Cl$ des diviseurs de Weil, le fibr\'e d\'eterminant admet
une racine: le fibr\'e pfaffien ${\cal{P}}$. De plus, pour $d\geq 4$ pair,
${\cal{L}}$ aussi admet une racine dans $\Cl$. Si $d=5$, le fibr\'e pfaffien
s'\'etend en un fibr\'e inversible et le groupe de Picard est engendr\'e par
${\cal{L}}$ et ${\cal{P}}$. On verra ainsi que la composante paire de
$\Theta_{\planp}(d)$ est localement factoriel si et seulement si $d=1,2,3$ ou
$d=5$. On obtient un r\'esultat analogue pour la composante impaire. Ensuite,
nous \'etudions la question d'existence d'une famille universelle sur un ouvert
de l'ouvert $U$ des th\^eta-caract\'eristiques dont le faisceau sous-jacent est
stable. Nous montrons que pour $d$ pair une telle famille ne peut exister,
tandis que pour $d$ impair une telle famille existe, pas sur $U$ entier, mais
localement dans la topologie de Zariski.


title:
Vector bundles on curves and generalized theta functions: recent results
  and open problems

abstract:
  Riemann surface carries a natural line bundle, the determinant bundle. The
space of sections of this line bundle (or its multiples) constitutes a natural
non-abelian generalization of the spaces of theta functions on the Jacobian.
There has been much progress in the last few years towards a better
understanding of these spaces, including a rigorous proof of the celebrated
Verlinde formula which gives their dimension. This survey paper tries to
explain what is now known and what remains open.


title: Nonnormal del Pezzo surfaces

abstract:
  This paper studies reduced, connected, Gorenstein surfaces with ample -K,
assumed to be reducible or nonnormal. The normalisation is a union of one or
more standard surfaces (scrolls and Veronese surfaces), marked with a conic as
double locus. The question is how to glue these together to get a Gorenstein
scheme. In characteristic 0, the results amount to a classification of
projective surfaces in the style of the 1880s. However, the methods involve a
study of the dualising sheaf of a nonnormal variety in terms of Rosenlicht
differentials, and there is a subtle pathology in characteristic p due to Mori
and S. Goto.


title: Bogomolov unstability on arithmetic surfaces

abstract:
  In this note, we will consider an arithmetic analogue of Bogomolov
unstability theorem.


title: Moduli of vector bundles on projective surfaces: some basic results

abstract:
  We prove that moduli spaces of torsion-free sheaves on a projective smooth
complex surface are irreducible, reduced and of the expected dimension,
provided the expected dimension is large enough. Actually we prove more: given
a line bundle on the surface, we show that the number of moduli of sheaves
which have a non-zero twisted (by the chosen line-bundle) endomorphism grows
slower than the expected dimension of the moduli space (for fixed rank and
increasing discriminant). The bounds we get are effective: we concentrate on
the case of rank two and we give a lower bound on the discriminant guaranteeing
that the moduli space is reduced of the expected dimension. We also give an
effective irreducibility result for complete intersections. All of the above
follows from a theorem bounding the dimension of complete subsets of the moduli
space which do not intersect the boundary.


title: On curves and surfaces with projectively equivalent hyperplane sections

abstract:
  In this paper we describe projective curves and surfaces such that almost all
their hyperplane sections are projectively equivalent. Our description is
complete for curves and close to being complete for smooth surfaces. In the
appendix we make some remarks on connections between the mentioned property of
a projective variety and its adjunction properties.


title: Degenerations of abelian surfaces and Hodge structures

abstract:
  In this paper the authors consider a certain toroidal compactification of the
moduli space of degenerations of (1,p)-polarized abelian surfaces with
(canonical) level structure. Using Hodge theory we give a proof that a
degenerate abelian surface associated to a corank 1 boundary point is (almost)
completely determined by the boundary point. The crucial ingredient in the
proof is the Local Invariant Cycle theorem which relates the variation of Hodge
structure to the mixed Hodge structure on the singular surface.


title: On the Existence of Supersingular Curves of Given Genus

abstract:
  We give a method to construct explicitly a supersingular curve of given genus
g in characteristic 2.


title: Crepant resolution of trihedral singularities

abstract:
  The purpose of this paper is to construct a crepant resolution of quotient
singularities by trihedral groups ( finite subgroups of SL(3,C) of certain type
), and prove that each Euler number of the minimal model is equal to the number
of conjugacy classes. Trihedral singularities are 3-dimensional version of
D_n-singularities, and they are non-isolated and many of them are not complete
intersections. The resolution is similar to one of D_n-singularities. There is
a nice combination of the toric resolution and Calabi-Yau resolution.


title: Moduli of abelian surfaces with a $(1,p^2)$ polarisation

abstract:
  We show that the (compactified) moduli space of abelian surfaces with a
polarisation of type $(1,p^2)$ is of general type for $p \ge 11$, improving a
result of O'Grady.


title: On complex surfaces diffeomorphic to rational surfaces

abstract:
  In this paper we prove that no complex surface of general type is
diffeomorphic to a rational surface, thereby completing the smooth
classification of rational surfaces and the proof of the Van de Ven conjecture
on the smooth invariance of Kodaira dimension.


title: Computing Multidimensional Residues

abstract:
  Given n polynomials in n variables with a finite number of complex roots, for
any of their roots there is a local residue operator assigning a complex number
to any polynomial. This is an algebraic, but generally not rational, function
of the coefficients. On the other hand, the global residue, which is the sum of
the local residues over all roots, depends rationally on the coefficients. This
paper deals with symbolic algorithms for evaluating that rational function.
Under the assumption that the deformation to the initial forms is flat, for
some choice of weights on the variables, we express the global residue as a
single residue integral with respect to the initial forms. When the input
equations are a Groebner basis, this leads to an efficient series expansion
algorithm for global residues, and to a vanishing theorem with respect to the
corresponding cone in the Groebner fan. The global residue of a polynomial
equals the highest coefficient of its (Groebner basis) normal form, and,
conversely, the entire normal form is expressed in terms of global residues.
This yields a method for evaluating traces over zero-dimensional complete
intersections. Applications include the counting of real roots, the computation
of the degree of a polynomial map, and the evaluation of multivariate symmetric
functions. All algorithms are illustrated for an explicit system in three
variables.


title:
Group cohomology construction of the cohomology of moduli spaces of flat
  connections on 2-manifolds

abstract:
  We use group cohomology and the de Rham complex on simplicial manifolds to
give explicit differential forms representing generators of the cohomology
rings of moduli spaces of representations of fundamental groups of 2-manifolds.
These generators are constructed using the de Rham representatives for the
cohomology of classifying spaces $BK$ where $K$ is a compact Lie group; such
representatives (universal characteristic classes) were found by Bott and
Shulman. Thus our representatives for the generators of the cohomology of
moduli spaces are given explicitly in terms of the Maurer-Cartan form. This
work solves a problem posed by Weinstein, who gave a corresponding construction
(following Karshon and Goldman) of the symplectic forms on these moduli spaces.
We also give a corresponding construction of equivariant differential forms on
the extended moduli space $X$, which is a finite dimensional symplectic space
equipped with a Hamiltonian action of $K$ for which the symplectic reduced
space is the moduli space of representations of the 2-manifold fundamental
group in $K$. (This paper is in press in Duke Math. J. The substance of the
text is unaltered; inor changes and corrections have been made to the file to
correspond to the version that will be published.)


title: Symplectic Forms on Moduli Spaces of Flat Connections on 2-manifolds

abstract:
  Let $G$ be a compact connected semisimple Lie group. We extend the techniques
of Weinstein [W] to give a construction in group cohomology of symplectic forms
$\omega$ on \lq twisted' moduli spaces of representations of the fundamental
group $\pi$ of a 2-manifold $\Sigma$ (the smooth analogues of ${\rm Hom}
(\pi_1(\Sigma), G)/G$) and on relative character varieties of fundamental
groups of 2-manifolds. We extend this construction to exhibit a symplectic form
on the extended moduli space [J1] (a Hamiltonian $G$-space from which these
moduli spaces may be obtained by symplectic reduction), and compute the moment
map for the action of $G$ on the extended moduli space.


title: Basic cohomology of associative algebras

abstract:
  We define a new cohomology for associative algebras which we compute for
algebras with units.


title: Conformal blocks, fusion rules and the Verlinde formula

abstract:
  The Verlinde formula computes the dimension of certain vector spaces
("conformal blocks") associated to a Rational Conformal Field Theory. In this
paper we show how this can be made rigorous for one particular such theory, the
WZW model. Thanks to the results of [B-L], [F] and [T-U-Y], this gives the
dimension of the space of global sections of the determinant line bundles (and
its multiples) on the moduli space of vector bundles with fixed rank and
determinant.


title: The blowup formula for Donaldson invariants

abstract:
  In this paper we present a formula which relates the Donaldson invariants of
a 4-manifold X with the Donaldson invariants of its blowup X#-CP(2). This
blow-up formula is independent of X and involves sigma-functions associated to
a naturally arising elliptic function. This, the final version, corrects some
earlier misconceptions regarding signs.


title: An Algorithmic Proof of Suslin's Stability Theorem over Polynomial Rings

abstract:
  Let $k$ be a field. Then Gaussian elimination over $k$ and the Euclidean
division algorithm for the univariate polynomial ring $k[x]$ allow us to write
any matrix in $SL_n(k)$ or $SL_n(k[x])$, $n\geq 2$, as a product of elementary
matrices. Suslin's stability theorem states that the same is true for the
multivariate polynomial ring $SL_n(k[x_1,\ldots ,x_m])$ with $n\geq 3$. As
Gaussian elimination gives us an algorithmic way of finding an explicit
factorization of the given matrix into elementary matrices over a field, we
develop a similar algorithm over polynomial rings.


title: Geometric invariant theory and flips

abstract:
  We study the dependence of geometric invariant theory quotients on the choice
of a linearization. We show that, in good cases, two such quotients are related
by a flip in the sense of Mori, and explain the relationship with the minimal
model programme. Moreover, we express the flip as the blow-up and blow-down of
specific ideal sheaves, leading, under certain hypotheses, to a quite explicit
description of the flip. We apply these ideas to various familiar moduli
problems, recovering results of Kirwan, Boden-Hu, Bertram-Daskalopoulos-
Wentworth, and the author. Along the way we display a chamber structure,
following Duistermaat-Heckman, on the space of all linearizations. We also give
a new, easy proof of the Bialynicki-Birula decomposition theorem.


title:
Higher-order differentials of the period map and higher Kodaira-Spencer
  classes

abstract:
  In \cite{K} we introduced two variants of higher-order differentials of the
period map and showed how to compute them for a variation of Hodge structure
that comes from a deformation of a compact K\"ahler manifold. More recently
there appeared several works (\cite{BG}, \cite{EV}, \cite{R}) defining higher
tangent spaces to the moduli and the corresponding higher Kodaira-Spencer
classes of a deformation. The $n^{th}$ such class $\kappa_n$ captures all
essential information about the deformation up to $n^{th}$ order. A well-known
result of Griffiths states that the (first) differential of the period map
depends only on the (first) Kodaira-Spencer class of the deformation. In this
paper we show that the second differential of the Archimedean period map
associated to a deformation is determined by $\kappa_2$ taken modulo the image
of $\kappa_1$, whereas the second differential of the usual period map, as well
as the second fundamental form of the VHS, depend only on $\kappa_1$ (Theorems
2, 5, and 6 in Section~3). Presumably, similar statements are valid in
higher-order cases (see Section~4).


title: A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

abstract:
  We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and
show that it maps polystable bundles to polystable ones. The role of ``dual''
variety to the given K3 surface $X$ is here played by a suitable component
$\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3
surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai
transform is invertible, and the image of a zero-degree stable bundle $F$ is
stable and has the same Euler characteristic as $F$.


title: On the A-D-E classification of the simple singularities of functions

abstract:
  We show how the classification of simple singularities of functions can be
reduced directly, not using the normal forms, to the classification of
irreducible Weyl groups. We also prove that the class of a singularity in its
local algebra always belongs to the tangent cone to the stratum $\mu =const$ of
the singularity. An interesting characterization of Coxeter elements among
quasi-Coxeter elements of a Weyl group (of one of the types A-D-E) is found.


title: Infinitesimal Deformations and Obstructions for Toric Singularities

abstract:
  The obstruction space T^2 and the cup product T^1 x T^1 -> T^2 are computed
for toric singularities.


title:
Some properties of Fano manifolds that are zeros of sections in
  homogenous vector bundles over Grassmannians

abstract:
  Let $X$ be a Fano manifold which is the zero scheme of a general global
section $s$ in an irreducible homogenous vector bundle over a Grassmannian. We
prove that the restriction of the Pl\"ucker embedding embeds $X$ projectively
normal, and that every small deformation of $X$ comes from a deformation of the
section $s$. These results are strengthened in the case of Fano 4-folds.