title:
An Application of a Log Version of the Kodaira Vanishing Theorem to
  Embedded Projective Varieties

abstract:
  Given an embedded smooth projective variety Y in CP^n, we show how the
existence of a hypersurface with high multiplicity along Y, but of relatively
low degree and log canonical near Y implies vanishing of higher cohomology for
certain twists of powers of the ideal sheaf of Y. This is applied to get rather
non-trivial vanishing in case Y is a universal determinantal variety (i.e.
quadratic Veronese, Segre embedding or Grassmannian of P^1's) or a curve
embedded by a complete linear series of high degree. The key in the
determinantal cases is to use the theory of complete objects. While the
theories of complete quadrics and linear maps yield the desired results, a
corresponding theory of complete skew forms seems not have been previously
worked out. The desired results are obtained in an appendix.


title: Stable pairs and log flips

abstract:
  This paper has two parts. In the first part, we review stable pairs and
triples on curves, leading up to Thaddeus' diagram of flips and contractions
starting from the blow-up of projective space along a curve embedded by a
complete linear series of the form K + ample. In the second part, we identify
log canonical divisors which exhibit Thaddeus' flips and contractions as "log"
flips and contractions in the sense of the log-minimal-model program.


title:
GKZ Systems, Gr\"obner Fans and Moduli Spaces of Calabi-Yau
  Hypersurfaces

abstract:
  We present a detailed analysis of the GKZ(Gel'fand, Kapranov and Zelevinski)
hypergeometric systems in the context of mirror symmetry of Calabi-Yau
hypersurfaces in toric varieties. As an application we will derive a concise
formula for the prepotential about large complex structure limits. (Talk given
at the Taniguchi Symposium ``Topological Field Theory, Primitive Forms and
Related Topics'' December, 1996)


title: Principal G-bundles over elliptic curves

abstract:
  Let $G$ be a simple and simply connected complex Lie group. We discuss the
moduli space of holomorphic semistable principal $G$ bundles over an elliptic
curve $E$. In particular we give a new proof of a theorem of Looijenga and
Bernshtein-Shvartsman, that the moduli space is a weighted projective space.
The method of proof is to study the deformations of certain unstable bundles
coming from special maximal parabolic subgroups of $G$. We also discuss the
associated automorphism sheaves and universal bundles, as well as the relation
between various universal bundles and spectral covers.


title: Irreducibility of \bar{M}_{0,n}(G/P,\beta)

abstract:
  Let G be a linear algebraic group, P be a parabolic subgroup of G and \beta
be a cycle of dimension 1 in the Chow group of the quotient G/P. Using
geometric arguments and Borel's fixed point theorem, we prove that the moduli
space \bar{M}_{0,n}(G/P, \beta) of n-pointed genus 0 stable maps representing
\beta is irreducible.


title: The intrinsic complexity of parametric elimination methods

abstract:
  This paper is devoted to the complexity analysis of a particular property,
called "algebraic robustness" owned by all known symbolic methods of parametric
polynomial equation solving (geometric elimination). It is shown that any
parametric elimination procedure which owns this property must neccessarily
have an exponential sequential time complexity.


title: Symplectic Deformations of Calabi-Yau threefolds

abstract:
  This manuscript from August 1995 (revised February 1996) studies the Kaehler
cone of Calabi-Yau threefolds via symplectic methods. For instance, it is shown
that if two Calabi-Yau threefolds are general in complex moduli and are
symplectic deformations of each other, then their Kaehler cones are the same.
The results are generalizations of those in the author's previous paper "The
Kaehler cone on Calabi-Yau threefolds" (Inventiones math. 107 (1992), 561-583;
Erratum: Inventiones math. 114 (1993), 231-233), where the behaviour of the
Kaehler cone under deformations of the complex structure was studied -- these
results may be recovered as a special case from this manuscript.
  The techniques used involve studying tamed almost complex deformations of the
complex structure on the Calabi-Yau threefold, and in particular proving the
non-vanishing of certain Gromov-Witten invariants, associated to codimension
one faces of the Kaehler cone. This in turn involves a detailed study of
primitive contractions on a smooth Calabi-Yau threefold, in particular for the
case of Type III contractions. More detailed information concerning
Gromov-Witten invariants associated to codimension one faces of the Kaehler
cone may be found in a recent preprint of the author.


title:
Flops, Type III contractions and Gromov-Witten invariants on Calabi-Yau
  threefolds

abstract:
  We investigate Gromov-Witten invariants associated to exceptional classes for
primitive birational contractions on a Calabi-Yau threefold X. It was observed
in a previous paper that these invariants are locally defined, in that they can
be calculated from knowledge of an open neighbourhood of the exceptional locus
of the contraction; in this paper, we make this explicit.
  For Type I contractions (i.e. only finitely many exceptional curves), a
method is given for calculating the Gromov-Witten invariants, and these in turn
yield explicit expressions for the changes in the cubic form $D^3$ and the
linear form $D.c_2$ under the corresponding flop. For Type III contractions
(when a divisor E is contracted to a smooth curve C of singularities), there
are only two relevant Gromov-Witten numbers n(1) and n(2). Here n(2) is the
number (suitably defined) of simple pseudo-holomorphic rational curves
representing the class of a fibre of E over C, and n(1) the number of simple
curves representing half this class. Explicit formulae for n(1) and n(2) are
given (n(1) in terms of the singular fibres of E over C and n(2)=2g(C)-2). An
easy proof of these formulae is provided when g(C)>0. The main part of the
paper then gives a proof valid in general (including the case g(C)=0).


title: Simplex, associahedron, and cyclohedron

abstract:
  The aim of the paper is to give an `elementary' introduction to the theory of
modules over operads and discuss three prominent examples of these objects -
simplex, associahedron (= the Stasheff polyhedron) and cyclohedron (= the
compactification of the space of configurations of points on the circle).
  Keywords: (right) module over an operad, module associated to a cyclic
operad, Koszul module over an operad.


title: Semiinfinite Flags. I. Case of global curve $P^1$

abstract:
  The Semiinfinite Flag Space appeared in the works of B.Feigin and E.Frenkel,
and under different disguises was found by V.Drinfeld and G.Lusztig in the
early 80-s. Another recent discovery (Beilinson-Drinfeld Grassmannian) turned
out to conceal a new incarnation of Semiinfinite Flags. We write down these and
other results scattered in folklore. We define the local semiinfinite flag
space attached to a semisimple group $G$ as the quotient $G((z))/HN((z))$ (an
ind-scheme), where $H$ and $N$ are a Cartan subgroup and the unipotent radical
of a Borel subgroup of $G$. The global semiinfinite flag space attached to a
smooth complete curve $C$ is a union of Quasimaps from $C$ to the flag variety
of $G$. In the present work we use $C=P^1$ to construct the category $PS$ of
certain collections of perverse sheaves on Quasimaps spaces, with factorization
isomorphisms. We construct an exact convolution functor from the category of
perverse sheaves on affine Grassmannian, constant along Iwahori orbits, to the
category $PS$. Conjecturally, this functor should correspond to the restriction
functor from modules over quantum group with divided powers to modules over the
small quantum group.


title: The tangent space at a special symplectic instanton bundle on P^{2n+1}

abstract:
  Let $MI_{Simp,P^{2n+1}}(k)$ be the moduli space of stable symplectic
instanton bundles on $P^{2n+1}$ with second Chern class $c_2=k$ (it is a closed
subscheme of the moduli space $MI_{P^{2n+1}}(k)$), We prove that the dimension
of its Zariski tangent space at a special (symplectic) instanton bundle is
$2k(5n-1)+4n^2-10n+3, k\geq 2$. It follows that special symplectic instanton
bundles are smooth points for $ k \leq 3 $


title: Weak semistable reduction in characteristic 0

abstract:
  Let X->B be a morphism of varieties in characteristic zero. Semistable
reduction has been proved for dim(B)=1 (Kempf, Knudsen, Mumford, Saint-Donat),
dim(X)=dim(B)-1 (de Jong) and dim(X)=dim(B)+2 (Alexeev, Kollar,
Shepherd-Barron). In this paper we consider the general case. First we define
what we mean by a semistable morphism in terms of toroidal embeddings. Then we
reduce the varieties to toroidal embeddings and solve a slightly weaker version
of semistable reduction. We also state the full semistable reduction problem in
terms of combinatorics of the associated polyhedral complexes.


title:
Rational and Non-Rational Algebraic Varieties: Lectures of J\'anos
  Koll\'ar

abstract:
  This is a detailed write-up of Koll\'ar's course at the EMS summer school in
Algebraic Geometry in Eger, Hungary, August 1996. The topics include
definitions and examples of rational and unirational varieties, with special
attention to varieties defined over non-algebraically closed fields, Segre's
theorem on non-rationality of cubic surfaces of Picard number one, Manin's
theorem that birationally equivalent cubic surfaces of Picard number one are
projectively equivalent, and Koll\'ar's method for constructing examples of low
degree non-rational hypersurfaces. Includes many exercises and their solutions.


title: Borel-Weil-Bott theory for Loop Groups

abstract:
  This note outlines the Borel-Weil-Bott theory for (arbitrary-genus) flag
varieties of loop groups, following ideas of G. Segal. Recent revivalist
interest in the "WZW fusion rules" suggests that my original (CMP 1995)
treatment of the matter is not all that well understood, so a brief account
thereof has been recycled along with the more recent results. Proofs are only
outlined, with full details given in the references.


title: Fano contact manifolds and nilpotent orbits

abstract:
  A contact structure on a complex manifold M is a corank 1 subbundle F of T(M)
such that the bilinear form on F with values in the quotient line bundle
L=T(M)/F deduced from the Lie bracket of vector fields is everywhere
non-degenerate. In this paper we consider the case where M is a Fano manifold;
this implies that L is ample. If g is a simple Lie algebra, the unique closed
orbit in P(g) (for the adjoint action) is a Fano contact manifold; it is
conjectured that every Fano contact manifold is obtained in this way. A
positive answer would imply an analogous result for compact quaternion-Kahler
manifolds with positive scalar curvature, a longstanding question in Riemannian
geometry. In this paper we solve the conjecture under the additional
assumptions that the group of contact automorphisms of M is reductive, and that
the image of the rational map M - - -> P(H0(M,L)*) associated to L has maximum
dimension. The proof relies on the properties of the nilpotent orbits in a
semi-simple Lie algebra, in particular on the work of R. Brylinski and B.
Kostant.


title: Rational Curves on Quasi-Projective Surfaces

abstract:
  The main result is that a quasi-projective surface has negative log Kodaira
dimension (i.e. no log pluricanonical sections) iff it is dominated by images
of the affine line. This follows from our main intermediate result, that the
smooth locus of a log Fano surface is rationally connected. We also give a
classification of all but a bounded family of rank one log del Pezzo surfaces.


title: New evidence for Green's conjecture on syzygies of canonical curves

abstract:
  We prove that two weakened forms of Green's conjectures for canonical curves
are equivalent when the genus $g$ is odd.


title: On Simplicial Commutative Rings with Vanishing Andr\'e-Quillen Homology

abstract:
  We propose a generalization of a conjecture of D. Quillen, on the vanishing
of Andr\'e-Quillen homology, to simplicial commutative rings. This conjecture
characterizes a notion of local complete intersection, extended to the
simplicial setting, under a suitable hypothesis on the local characteristic.
Further, under the condition of finite-type homology, we then prove the
conjecture in the case of a simplicial commutative algebra augmented over a
field of non-zero characteristic. As a consequence, we obtain a proof of
Quillen's conjecture for a Noetherian commutative algebra - again augmented
over a field of non-zero characteristic.


title: Localization of virtual classes

abstract:
  We prove a localization formula for virtual fundamental classes in the
context of torus equivariant perfect obstruction theories. As an application,
the higher genus Gromov-Witten invariants of projective space are expressed as
graph sums of tautological integrals over moduli spaces of stable pointed
curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals
over spaces of higher genus multiple covers are computed.


title: Discriminant Complements and Kernels of Monodromy Representations

abstract:
  We show that the kernel of the monodromy representation for hypersurfaces of
degree d and dimension n is large for d at least three with the exception of
the cases (d,n) = (3,0) and (3,1). For these the kernel is finite. By "large"
we mean a group that admits a homomorphism to a semisimple Lie group of
noncompact type with Zariski-dense image. By the Tits alternative a large group
contains a free subgroup of rank two.


title: Singular hermitian metrics on vector bundles

abstract:
  We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphic
vector bundles and define positivity in view of $L^2$-estimates. Associated
with a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of a
certain $d''$-complex. We prove a vanishing theorem for the cohomology of this
sheaf. All this generalizes to the case of higher rank known results of Nadel
for the case of line bundles. We introduce a new semi-positivity notion,
$t$-nefness, for vector bundles, establish some of its basic properties and
prove that on curves it coincides with ordinary nefness. We particularize the
results on s.h.m. to the case of vector bundles of the form $E=F \otimes L$,
where $F$ is a $t$-nef vector bundle and $L$ is a positive (in the sense of
currents) line bundle. As applications we generalize to the higher rank case 1)
Kawamata-Viehweg Vanishing Theorem, 2) the effective results concerning the
global generation of jets for the adjoint to powers of ample line bundles, and
3) Matsusaka Big Theorem made effective.


title: Effective nonvanishing, effective global generation

abstract:
  We prove a multiple-points higher-jets nonvanishing theorem by the use of
local Seshadri constants. Applications are given to effectivity problems such
as constructing rational and birational maps into Grassmannians, and the global
generation of vector bundles.


title: A remark on virtual orientations for complete intersections

abstract:
  The aim of this note is to give a simple definition of genus zero virtual
orientation classes (or fundamental classes) for projective complete
intersections or, more generally, for complete intersections in convex
varieties, and to prove a push forward formula for them.


title: Duality and flat base change on formal schemes

abstract:
  We give several related versions of global Grothendieck Duality for unbounded
complexes on noetherian formal schemes. The proofs, based on a non-trivial
adaptation of Deligne's method for the special case of ordinary schemes, are
reasonably self-contained, modulo the Special Adjoint Functor Theorem. An
alternative approach, inspired by Neeman and based on recent results about
"Brown Representability," is indicated as well. A section on applications and
examples illustrates how these theorems synthesize a number of different
duality-related results (local duality, formal duality, residue theorems,
dualizing complexes...).
  A flat-base-change theorem for pseudo-proper maps leads in particular to
sheafified versions of duality for bounded-below complexes with quasi-coherent
homology. Thanks to Greenlees-May duality, the results take a specially nice
form for proper maps and bounded-below complexes with coherent homology.


title: Torsion of abelian varieties, Weil classes and cyclotomic extensions

abstract:
  Let $K$ be a field finitely generated over the field of rational numbers,
$K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an
infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We
prove that under certain conditions on $X$ and $K$ the existence of infinitely
many L-rational points of finite order on $X$ implies that the intersection of
$L$ and $K(c)$ has infinite degree over $K$.


title: Local structure of moduli spaces

abstract:
  We describe the algebra of a universal formal deformation as the zeroth
cohomology of the dg Lie algebra corresponding to this deformation problem. A
report at Arbeitstagung 1997 on the joint work with V.Hinich.


title: Desingularized moduli spaces of sheaves on a K3, I

abstract:
  Moduli spaces of semistable torsion-free sheaves on a K3 surface $X$ are
often holomorphic symplectic varieties, deformation equivalent to a Hilbert
scheme parametrizing zero-dimensional subschemes of $X$. In fact this should
hold whenever semistability is equivalent to stability. In this paper we study
a typical "opposite" case, i.e. the moduli space $M_c$ of semistable rank-two
torsion-free sheaves on $X$ with trivial determinant and second Chern class
equal to an even number $c$. The moduli space $M_c$ always contains points
corresponding to strictly semistable sheaves. If $c$ is at least 4, then $M_c$
is singular along the locus parametrizing strictly semistable sheaves, and on
the smooth locus of $M_c$ there is a symplectic holomorphic form. Thus it is
natural to ask whether there is a symplectic desingularization of $M_c$. We
construct such a desingularization for $c=4$; in another paper we show that
this desingularization gives a new deformation class of (K\"ahler) holomorphic
irreducible symplectic varieties (of dimension ten). We also study the case
$c>4$. We describe what should be an interesting desingularization, however we
are not able to produce a symplectic one. In fact we suspect there is no
symplectic smooth model of $M_c$ if $c>4$ (and even, of course).


title: Limits in $n$-categories

abstract:
  We define notions of direct and inverse limits in an $n$-category. We prove
that the $n+1$-category $nCAT'$ of fibrant $n$-categories admits direct and
inverse limits. At the end we speculate (without proofs) on some applications
of the notion of limit, including homotopy fiber product and homotopy coproduct
for $n$-categories, the notion of $n$-stack, representable functors, and
finally on a somewhat different note, a notion of relative Malcev completion of
the higher homotopy at a representation of the fundamental group.


title: Some finiteness results for Calabi-Yau threefolds

abstract:
  We investigate the moduli theory of Calabi--Yau threefolds, and using
Griffiths' work on the period map, we derive some finiteness results. In
particular, we confirm a prediction of Morrison's Cone Conjecture.


title:
Euler number of the compactified Jacobian and multiplicity of rational
  curves

abstract:
  We show that the Euler number of the compactified Jacobian of a rational
curve $C$ with locally planar singularities is equal to the multiplicity of the
$\delta$-constant stratum in the base of a semi-universal deformation of $C$.
In particular, the multiplicity assigned by Yau, Zaslow and Beauville to a
rational curve on a K3 surface $S$ coincides with the multiplicity of the
normalisation map in the moduli space of stable maps to $S$.


title: Bend, break and count II : elliptics, cuspidals, linear genera

abstract:
  Extending the method of Part I (alg-geom/9704004), we give recursive formulae
for: the genus-1 Severi degree (formula first found by Getzler), the degree of
the variety of 1-cuspidal curves of genus 0 or 1, and the linear (sectional)
geometric genus of the genus-0 and genus-1 Severi varieties. The proof is short
and elementary.


title: A stratification of the moduli space of vector bundles on curves

abstract:
  Let $E$ be a vector bundle of rank $r\geq 2$ on a smooth projective curve $C$
of genus $g \geq 2$ over an algebraically closed field $K$ of arbitrary
characteristic. For any integer with $1\le k\le r-1$ we define
$${\se}_k(E):=k\deg E-r\max\deg F.$$ where the maximum is taken over all
subbundles $F$ of rank $k$ of $E$. The ${s}_k$ gives a stratification of the
moduli space ${\cal M}(r,d)$ of stable vector bundles of rank $r$ and degree on
$d$ on $C$ into locally closed subsets ${\calM}(r,d,k,s)$ according to the
value of $s$ and $k$. There is a component ${\cal M}^0(r,d,k,s)$ of ${\cal
M}(r,d,k,s)$ distinguish by the fact that a general $E\in {\cal M}^0(r,d,k,s)$
admits a stable subbundle $F$ such that $E/F$ is also stable. We prove: {\it
For $g\ge \frac{r+1}{2}$ and $0<s\leq k(r-k)(g-1) +(r+1)$, $s\equiv kd \mod r,$
${\cal M}^0(r,d,k,s)$ is non-empty,and its component ${\cal M}^0(r,d,k,s)$ is
of dimension} $$\dim {\cal M}^0(r,d,k,s)=\left\{\begin{array}{lcl}
(r^2+k^2-rk)(g-1)+s-1& &s<k(r-k)(g-1) &{\rm if}& r^2(g-1)+1& & s\ge
k(r-k)(g-1)\end{array}\right.$$


title: Exact Random Walk Distributions using Noncommutative Geometry

abstract:
  Using the results obtained by the non commutative geometry techniques applied
to the Harper equation, we derive the areas distribution of random walks of
length $ N $ on a two-dimensional square lattice for large $ N $, taking into
account finite size contributions.


title: Nef Divisors on Moduli Spaces of Abelian Varieties

abstract:
  We determine the cone of nef divisors on the Voronoi compactification A_g^*
of the moduli space A_g of principally polarized abelian varieties of dimension
g for genus g=2,3. As a corollary we obtain that the spaces A_g^*(n) with
level-n structure are a minimal, resp. canonical, model for g=2, n>=4, resp.
n>=5 and g=3, n>=3, resp. n>=4. We give two proofs: The easy and quick one
reduces the problem to \bar M_g where we can use a result of Faber. This
approach cannot be generalized to higher genus g. The main point of the paper
is, therefore, to give a second proof using theta functions and a result of
Weissauer. This technique can be at least partially generalized to higher
genus. We formulate a conjecture for the nef cone of A_g^* for all g.


title: On algebra generated by Chern-Bott forms on SL_n/B

abstract:
  In this short note we give an explicit presentation of the algebra A_n
generated by the curvature 2-forms of the standard Hermitiam line bundles over
SL_n/B as the quotient of the polynomial ring. The difference between A_n and
H^*(SL_n/B) reflects the fact that SL_n/B is not a symmetric space. Possible
applications of A_n lie in the field of arithmetic intersection theory on flag
varieties.


title: Conormal Geometry of Maximal Minors

abstract:
  Let A be a Noetherian local domain, N be a finitely generated torsion- free
module, and M a proper submodule that is generically equal to N. Let A[N] be an
arbitrary graded overdomain of A generated as an A-algebra by N placed in
degree 1. Let A[M] be the subalgebra generated by M. Set C:=Proj(A[M]) and
r:=dim C. Form the (closed) subset W of Spec(A) of primes p where A[N]_p is not
a finitely generated module over A[M]_p, and denote the preimage of W in C by
E. We prove this: (1) dim E=r-1 if either (a) N is free and A[N] is the
symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2)
E is equidimensional if (a) holds and A is universally catenary.
  Our proof was inspired by some recent work of Gaffney and Massey, which we
sketch; they proved (2) when A is the ring of germs of a complex- analytic
variety, and applied it to perfect a characterization of Thom's A_f-condition
in equisingularity theory. From (1), we recover, with new proofs, the usual
height inequality for maximal minors and an extension of it obtained by the
authors in 1992. From the latter, we recover the authors' generalization to
modules of B"oger's criterion for integral dependence of ideals. Finally, we
introduce an application of (1), being made by the second author, to the
geometry of the dual variety of a projective variety, and use it to obtain an
interesting example where the conclusion of (1) fails and A[N] is a finitely
generated module over A[M].


title: Stability of the Rational Homotopy Type of Moduli Spaces

abstract:
  We show that for g > 2k+2 the k-rational homotopy type of the moduli space
M_{g,n} of algebraic curves of genus g with n punctures is independent of g,
and the space M_{g,n} is k-formal. This implies the existence of a limiting
rational homotopy type of M_{g,n} as g goes to infinity and the formality of
it.


title: A property deducible from the generic initial ideal

abstract:
  Let $S_d$ be the vector space of monomials of degree $d$ in the variables
$x_1, ..., x_s$. For a subspace $V \sus S_d$ which is in general coordinates,
consider the subspace $\gin V \sus S_d$ generated by initial monomials of
polynomials in $V$ for the revlex order. We address the question of what
properties of $V$ may be deduced from $\gin V$. % This is an approach for
understanding what algebraic or geometric properties of a homogeneous ideal $I
\sus k[x_1, ..., x_s]$ that may be deduced from its generic initial ideal $\gin
I$.


title: Determinantal schemes and Buchsbaum-Rim sheaves

abstract:
  Let $\phi$ be a generically surjective morphism between direct sums of line
bundles on $\proj{n}$ and assume that the degeneracy locus, $X$, of $\phi$ has
the expected codimension. We call $B_{\phi} = \ker \phi$ a (first)
Buchsbaum-Rim sheaf and we call $X$ a standard determinantal scheme. Viewing
$\phi$ as a matrix (after choosing bases), we say that $X$ is good if one can
delete a generalized row from $\phi$ and have the maximal minors of the
resulting submatrix define a scheme of the expected codimension. In this paper
we give several characterizations of good determinantal schemes. In particular,
it is shown that being a good determinantal scheme of codimension $r+1$ is
equivalent to being the zero-locus of a regular section of the dual of a first
Buchsbaum-Rim sheaf of rank $r+1$. It is also equivalent to being standard
determinantal and locally a complete intersection outside a subscheme $Y
\subset X$ of codimension $r+2$. Furthermore, for any good determinantal
subscheme $X$ of codimension $r+1$ there is a good determinantal subscheme $S$
codimension $r$ such that $X$ sits in $S$ in a nice way. This leads to several
generalizations of a theorem of Kreuzer. For example, we show that for a
zeroscheme $X$ in $\proj{3}$, being good determinantal is equivalent to the
existence of an arithmetically Cohen-Macaulay curve $S$, which is a local
complete intersection, such that $X$ is a subcanonical Cartier divisor on $S$.


title: Buchsbaum-Rim sheaves and their multiple sections

abstract:
  This paper begins by introducing and characterizing Buchsbaum-Rim sheaves on
$Z = \Proj R$ where $R$ is a graded Gorenstein K-algebra. They are reflexive
sheaves arising as the sheafification of kernels of sufficiently general maps
between free R-modules. Then we study multiple sections of a Buchsbaum-Rim
sheaf $\cBf$, i.e, we consider morphisms $\psi: \cP \to \cBf$ of sheaves on $Z$
dropping rank in the expected codimension, where $H^0_*(Z,\cP)$ is a free
R-module. The main purpose of this paper is to study properties of schemes
associated to the degeneracy locus $S$ of $\psi$. It turns out that $S$ is
often not equidimensional. Let $X$ denote the top-dimensional part of $S$. In
this paper we measure the ``difference'' between $X$ and $S$, compute their
cohomology modules and describe ring-theoretic properties of their coordinate
rings. Moreover, we produce graded free resolutions of $X$ (and $S$) which are
in general minimal. Among the applications we show how one can embed a
subscheme into an arithmetically Gorenstein subscheme of the same dimension and
prove that zero-loci of sections of the dual of a null correlation bundle are
arithmetically Buchsbaum.


title:
An elliptic conic bundle in P^4 arising from a stable rank-3 vector
  bundle

abstract:
  In this note we show the existence of a family of elliptic conic bundles in
P^4 of degree 8. This family has been overlooked and in fact falsely ruled out
in a series of classification papers. Our surfaces provide a counterexample to
a conjecture of Ellingsrud and Peskine. According to this conjecture there
should be no irregular m-ruled surface in P^4 for m at least 2.


title:
Relations between the correlators of the topological sigma-model coupled
  to gravity

abstract:
  We prove a new recursive relation between the correlators $<
\tau_{d_1}\gamma_1...\tau_{d_n}\gamma_n >_{g,\beta}$, which together with known
relations allows one to express all of them through the full system of
Gromov-Witten invariants in the sense of Kontsevich-Manin and the intersection
indices of tautological classes on $\bar{M}_{g,n}$, effectively calculable in
view of earlier results due to Mumford, Kontsevich, Getzler, and Faber. This
relation shows that a linear change of coordinates of the big phase space
transforms the potential with gravitational descendants to another function
defined completely in terms of the Gromov-Witten correspondence and the
intersection theory on $V^n\times\bar{M}_{g,n}$. We then extend the formalism
of gravitational descendants from quantum cohomology to more general Frobenius
manifolds and Cohomological Field Theories.


title: Duality Construction of Moduli Spaces

abstract:
  We show for the moduli space of rank-2 coherent sheaves on an algebraic
surface that there exists a 'dual' moduli space. This dual space allows a
construction of the first one without using the GIT construction. Furthermore,
we obtain a Barth-morphism, generalizing the concept of jumping lines. This
morphism is by construction a finite morphism.


title:
Pieri-type formulas for maximal isotropic Grassmannians via triple
  intersections

abstract:
  We give an elementary proof of the Pieri-type formula in the cohomology of a
Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic
vector space. This proof proceeds by explicitly computing a triple intersection
of Schubert varieties. The decisive step is an explicit description of the
intersection of two Schubert varieties, from which the multiplicities (which
are powers of 2) in the Pieri-type formula are deduced.


title: Rational cuspidal plane curves of type (d, d-3)

abstract:
  In the previous paper [E-print alg-geom/9507004] we classified the rational
cuspidal plane curves C with a cusp of multiplicity deg C - 2. In particular,
we showed that any such curve can be transformed into a line by Cremona
transformations. Here we do the same for the rational cuspidal plane curves C
with a cusp of multiplicity deg C - 3.


title: Wall-crossing formulae for algebraic surfaces with $q>0$

abstract:
  We extend the ideas of Friedman and Qin (Flips of moduli spaces and
transition formulae for Donaldson polynomial invariants of rational surfaces)
to find the wall-crossing formulae for the Donaldson invariants of algebraic
surfaces with geometrical genus zero, positive irregularity and anticanonical
divisor effective, for any wall $\zeta$ with $l_{\zeta}=(\zeta\sp{2}-p_1)/4$
being zero or one.


title: Counting curves of any genus on rational ruled surfaces

abstract:
  In this paper we study the geometry of the Severi varieties parametrizing
curves on the rational ruled surface $\fn$. We compute the number of such
curves through the appropriate number of fixed general points on $\fn$, and the
number of such curves which are irreducible. These numbers are known as Severi
degrees; they are the degrees of unions of components of the Hilbert scheme. As
(i) $\fn$ can be deformed to $\eff_{n+2}$, (ii) the Gromov-Witten invariants
are deformation-invariant, and (iii) the Gromov-Witten invariants of $\eff_0$
and $\eff_1$ are enumerative, Theorem \ref{irecursion} computes the genus $g$
Gromov-Witten invariants of all $\fn$. (The genus 0 case is well-known.) The
arguments are given in sufficient generality to also count plane curves in the
style of L. Caporaso and J. Harris and to lay the groundwork for computing
higher genus Gromov-Witten invariants of blow-ups of the plane at up to five
points (in a future paper).


title:
Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane
  curves with fixed multiple points

abstract:
  As another application of the degeneration methods of [V3], we count the
number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed
multiple points on a conic $E$, not containing $E$, through an appropriate
number of general points in the plane. As a special case, we count the number
of irreducible genus $g$ curves in any divisor class $D$ on the blow-up of the
plane at up to five points (no three collinear). We then show that these
numbers give the genus $g$ Gromov-Witten invariants of the surface. Finally, we
suggest a direction from which the remaining del Pezzo surfaces can be
approached, and give a conjectural algorithm to compute the genus g
Gromov-Witten invariants of the cubic surface.


title: Very ampleness for Theta on the compactified Jacobian

abstract:
  The Jacobian $J$ of a complete, smooth, connected curve $X$ admits a
canonical divisor $\Theta$, called the Theta divisor. It is well-known that
$\Theta$ is ample and, in fact, $3\Theta$ is very ample. For a general
complete, integral curve $X$, D'Souza constructed a compactification $\bar J$
of the Jacobian $J$ by considering torsion-free, rank 1 sheaves on $X$.
Soucaris and the author considered independently the analogous Theta divisor
$\Theta$ on $\bar J$, and showed that $\Theta$ is ample. In this article, we
show that $n\Theta$ is very ample for $n$ greater or equal to a specified lower
bound. If $X$ has at most ordinary nodes or cusps as singularities, then our
lower bound is 3. Our main tool is to use theta sections associated to vector
bundles on $X$ to embed $\bar J$ into a projective space.


title: Product formulas for modular forms on O(2,n) (after R.Borcherds)

abstract:
  This is my talk on the Bourbaki seminar, November 1996. It contains an
elementary introduction to Borcherds' product formulas.


title:
The enumerative geometry of rational and elliptic curves in projective
  space

abstract:
  We study the geometry of varieties parametrizing degree d rational and
elliptic curves in P^n intersecting fixed general linear spaces and tangent to
a fixed hyperplane H with fixed multiplicities along fixed general linear
subspaces of H. As an application, we derive recursive formulas for the number
of such curves when the number is finite. These recursive formulas require as
``seed data'' only one input: there is one line in P^1 through two points.
These numbers can be seen as top intersection products of various cycles on the
Hilbert scheme of degree d rational or elliptic curves in P^n, or on certain
components of $\mbar_0(P^n,d)$ or $\mbar_1(P^n,d)$, and as such give
information about the Chow ring (and hence the topology) of these objects. The
formula can also be interpreted as an equality in the Chow ring (not
necessarily at the top level) of the appropriate Hilbert scheme or space of
stable maps. In particular, this gives an algorithm for counting rational and
elliptic curves in P^n intersecting various fixed general linear spaces. (The
genus 0 numbers were found earlier by Kontsevich-Manin, and the genus 1 numbers
were found for n=2 by Ran and Caporaso-Harris, and independently by Getzler for
n=3.)


title: Separation properties of theta functions

abstract:
  In a 1993 article, G. Faltings gave a new construction of the moduli space
$U$ of semistable vector bundles on a smooth curve $X$, avoiding geometric
invariant theory. Roughly speaking, Faltings showed that the normalisation $B$
of the ring $A$ of theta functions (associated with vector bundles on $X$)
suffices to realize $U$ as a projective variety. Describing Faltings' work,
C.S. Seshadri asked how close $A$ is to $B$. In this article, we address this
question from a geometric point of view. We consider the rational map, $\pi : U
@>>> Proj(A)$, and show that, not only is $\pi$ defined everywhere, but also
$\pi$ is bijective, and is an isomorphism over the stable locus of $U$, if the
characteristic of the ground field is 0. Moreover, we give a direct local
construction of $U$ as a fine moduli space, when the rank and degree are
coprime, in any characteristic. The methods in the article apply to singular
curves as well.


title: Compactifying the relative Jacobian over families of reduced curves

abstract:
  We construct natural relative compactifications for the relative Jacobian
over a family $X/S$ of reduced curves. In contrast with all the available
compactifications so far, ours admit a universal sheaf, after an etale base
change. Our method consists of considering the functor $F$ of relatively
simple, torsion-free, rank 1 sheaves on $X/S$, and showing that certain open
subsheaves of $F$ have good properties. Strictly speaking, the functor $F$ is
only representable by an algebraic space, but we show that $F$ is representable
by a scheme after an etale base change. Finally, we use theta functions
originating from vector bundles to compare our new compactifications with the
available ones.


title: Height zeta functions of toric bundles over flag varieties

abstract:
  We investigate analytic properties of height zeta functions of toric bundles
over flag varieties.


title: Local-global intersection homology

abstract:
  This paper defines new intersection homology groups. The basic idea is this.
Ordinary homology is locally trivial. Intersection homology is not. It may have
significant local cycles. A local-global cycle is defined to be a family of
such local cycles that is, at the same time, a global cycle. The motivating
problem is the numerical characterisation of the flag vectors of convex
polytopes. Central is a study of the cycles on a cone and a cylinder, in terms
of those on the base. This leads to the topological definition of local-global
intersection homology, and a formula for the expected Betti numbers of toric
varieties. Various related questions are also discussed.


title:
The Pontryagin rings of moduli spaces of arbitrary rank holomorphic
  bundles over a Riemann surface

abstract:
  The cohomology of the moduli spaces of stable bundles M(n,d), of coprime rank
n and degree d, over a Riemann surface (of genus g > 1) have been intensely
studied over the past three decades. We prove in this paper that the Pontryagin
ring of M(n,d) vanishes in degrees above 2n(n-1)(g-1) and that this bound is
strict (i.e. there exists a non-zero element of degree 2n(n-1)(g-1) in
Pont(M(n,d)).) This result is a generalisation of a 1967 Newstead conjecture
that Pont(M(2,1)) vanished above 4(g-1) (or equivalently that \beta^g =0.)
These results have been independently proved by Lisa Jeffrey and Jonathan
Weitsman.


title: On Weierstrass points and optimal curves

abstract:
  We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness
(up to isomorphism) of some optimal curves.


title: Fibr\'es prioritaires g\'en\'eriques instables sur le plan projectif

abstract:
  The structure of the generic prioritary sheaf on the projective plane is
given, when it cannot be semi-stable


title: Multiplier Ideals, Vanishing Theorem and Applications

abstract:
  The purpose of this note is to give a survey of the algebraic properties of
multiplier ideals, and illustrate some of their applications to classical
projective geometry.


title: A Complex Hyperbolic Structure for Moduli of Cubic Surfaces

abstract:
  We show that the moduli space M of marked cubic surfaces is biholomorphic to
the quotient by a discrete group generated by complex reflections of the
complex four-ball minus the reflection hyperplanes of the group. Thus M carries
a complex hyperbolic structure: an (incomplete) metric of constant holomorphic
sectional curvature.


title: The Determinant of a Hypergeometric Period Matrix

abstract:
  We consider a function $U=e^{-f_0}\prod_j^N f_j^{\alpha_j}$ on a real affine
space, here $f_0,..,f_N$ are linear functions, $\alpha_1, ...,\alpha_N$ complex
numbers. The zeros of the functions $f_1, ..., f_N$ form an arrangement of
hyperplanes in the affine space. We study the period matrix of the
hypergeometric integrals associated with the arrangement and the function $U$
and compute its determinant as an alternating product of gamma functions and
critical points of the functions $f_0,..., f_N$ with respect to the
arrangement. In the simplest example, $N=1, f_0=f_1=t$, the determinant formula
takes the form $\int_0^\infty e^{-t} t^{\alpha -1} dt=\Gamma (\alpha).$
  We also give a determinant formula for Selberg type exponential integrals. In
this case the arangements of hyperplanes is special and admits a symmetry
group, the period matrix is decomposed into blocks corresponding to different
representations of the symmetry group on the space of the hypergeometric
integrals associated with the arrangement. We compute the determinant of the
block corresponding to the trivial representation.


title: Degeneracy Schemes and Schubert Varieties

abstract:
  A result of Zelevinsky states that an orbit closure in the space of
representations of the equioriented quiver of type $A_h$ is in bijection with
the opposite cell in a Schubert variety of a partial flag variety $SL(n)/Q$. We
prove that Zelevinsky's bijection is a scheme-theoretic isomorphism, which
shows that the universal degeneracy schemes of Fulton are reduced and
Cohen-Macaulay in arbitrary characteristic.


title: Kawachi's invariant for normal surface singularities

abstract:
  We study a useful numerical invariant of normal surface singularities,
introduced recently by T. Kawachi. Using this invariant, we give a quick proof
of the (well-known) fact that all log-canonical surface singularities are
either elliptic Gorenstein or rational (without assuming a priori that they are
Q-Gorenstein).
  The main result is a criterion (stated in terms of Kawachi's invariant)
regarding global generation of adjoint linear systems on normal surfaces with
boundary. Such results can be used in proving effective estimates for global
generation on singular threefolds. The theorem of Ein-Lazarsfeld and Kawamata,
which says that the minimal center of log-canonical singularities is always
normal, explains why the results proved here are relevant in that situation.


title:
Very ampleness of adjoint linear systems on smooth surfaces with
  boundary

abstract:
  Let M be a Q-divisor on a smooth surface over C. In this paper we give
criteria for very ampleness of the adjoint of the round-up of M. (Similar
results for global generation were given by Ein and Lazarsfeld and used in
their proof of Fujita's Conjecture in dimension 3.) In the last section we
discuss an example which suggests that this kind of criteria might also be
useful in the study of linear systems on surfaces.


title: The family G_T of graded quotients of k[x,y] of given Hilbert function

abstract:
  The nonsingular variety G_T parametrizes all graded ideals I of R=k[x,y] for
which the Hilbert function H(R/I)=T. The variety G_T has a natural cellular
decomposition: each cell V(E) corresponds to a monomial ideal E for which
H(R/E)=T. Given a monomial ideal E, the quotient R/E has a basis of monomials
in the shape of a partition P(E), having "diagonal lengths" T. The dimension of
the cell V(E) is the number of difference one hooks of P(E).
  We show that G_T is birational to a certain product SGrass(T) of small
Grassman varieties, and that over the complexes, the birational map induces an
isomorphism of homology groups, but not usually an isomorphism of rings. We
determine the homology ring) when T=(1,2,...,d-1,d,...,d,1). This G(T) is the
desingularisation of the d-secant bundle Sec(d,j) of the degree-j rational
normal curve, We determine the classes of the pullbacks of the higher singular
loci of Sec(d,j). We also use the ring H*(G_T) to determine the number of
ideals satisfying an intersection of ramification conditions at different
points of P^1.
  A main tool is a combinatorial "hook code" for the partition P(E), that gives
the image of the cell V(E) in SGrass(T).


title: Wall-Crossing functors and D-modules

abstract:
  We study Translation functors and Wall-Crossing functors on infinite
dimensional representations of a complex semisimple Lie algebra using
D-modules. This functorial machinery is then used to prove the
Endomorphism-theorem and the Structure-theorem, two important results
established earlier by W. Soergel in a totally different way. Other
applications to the category O of Bernstein-Gelfand-Gelfand are given, and some
conjectural relationships between Koszul duality, Verdier duality and
convolution functors are discussed. A geometric interpretation of tilting
modules is given.


title: A generalization of curve genus for ample vector bundles, II

abstract:
  Let $X$ be a compact complex manifold of dimension $n\ge 2$ and $\ce$ an
ample vector bundle of rank $r<n$ on $X$. As the continuation of Part I, we
further study the properties of $g(X,\ce)$ that is an invariant for pairs
$(X,\ce)$ and is equal to curve genus when $r=n-1$. Main results are the
classifications of $(X,\ce)$ with $g(X,\ce)=2$ (resp. 3) when $\ce$ has a
regular section (resp. $\ce$ is ample and spanned) and $1<r<n-1$.


title: A generalization of curve genus for ample vector bundles, I

abstract:
  A new genus $g=g(X,\ce)$ is defined for the pairs $(X,\ce)$ that consist of
$n$-dimensional compact complex manifolds $X$ and ample vector bundles $\ce$ of
rank $r$ less than $n$ on $X$. In case $r=n-1$, $g$ is equal to curve genus.
Above pairs $(X,\ce)$ with $g$ less than two are classified. For spanned $\ce$
it is shown that $g$ is greater than or equal to the irregularity of $X$, and
its equality condition is given.


title: On Moduli of G-bundles over Curves for exceptional G

abstract:
  Let $G$ be a simple and simply connected complex Lie group, ${\goth{g}}$ its
Lie algebra. I remove the restriction ``$G$ is of classical type or $G_2$''
made on $G$ in the papers of Beauville, Laszlo and myself [L-S] and [B-L-S] on
the moduli of principal G-bundles over a curve. As I will just "patch" the
missing technical points, this note should be seen as an appendix to the above
cited papers.


title: On the width of lattice-free simplices

abstract:
  Among integral polytopes (vertices with integral coordinates), lattice-free
polytopes - intersecting the lattice ONLY at their vertices- are of particular
interestin combinatorics and geometry of numbers.
  A natural question is to measure their "width" (with respect to the integral
lattice).There were no known examples of lattice-free polytopes with width
bigger than 2 .We prove the following Theorem : Given any positive number
$\alpha$ strictly inferior to $1/e$, for d large enough there exists a
lattice-free simplex of dimension d and width superior to $\alpha d$.


title: On Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3 folds

abstract:
  In this paper, we verify a part of the Mirror Symmetry Conjecture for
Schoen's Calabi-Yau 3-fold, which is a special complete intersection in a toric
variety. We calculate a part of the prepotential of the A-model Yukawa
couplings of the Calabi-Yau 3-fold directly by means of a theta function and
Dedekind's eta function. This gives infinitely many Gromov-Witten invariants,
and equivalently infinitely many sets of rational curves in the Calabi-Yau
3-fold. Using the toric mirror construction, we also calculate the prepotential
of the B-model Yukawa couplings of the mirror partner. Comparing the expansion
of the B-model prepotential with that of the A-model prepotential, we check a
part of the Mirror Symmetry Conjecture up to a high order.


title: Resolution of Singularities

abstract:
  This article is an exposition of an elementary constructive proof of
canonical resolution of singularities in characteristic zero, presented in
detail in Invent. Math. 128 (1997), 207-302. We define a new local invariant
and get an algorithm for canonical desingularization by successively blowing up
its maximum loci. The invariant can be described by local computations that
provide equations for the centres of blowing up. We describe the origin of our
approach and present the proof (in the hypersurface case) in parallel with a
worked example.


title: Vector Bundles over Elliptic Fibrations

abstract:
  This paper gives various methods for constructing vector bundles over
elliptic curves and more generally over families of elliptic curves. We
construct universal families over generalized elliptic curves via spectral
cover methods and also by extensions, and then give a relative version of the
construction in families. We give various examples and make Chern class
computations.


title: A Survey of the Hodge Conjecture for Abelian Varieties

abstract:
  We review what is known about the Hodge conjecture for abelian varieties,
with some emphasis on how Mumford-Tate groups have been applied to this
problem.


title: K3 Surfaces with Nine Cusps

abstract:
  By a K3-surface with nine cusps I mean a surface with nine isolated double
points A_2, but otherwise smooth, such that its minimal desingularisation is a
K3-surface. It is shown, that such a surface admits a cyclic triple cover
branched precisely over the cusps. This parallels the theorem of Nikulin, that
a K3-surface with 16 nodes is a Kummer quotient of a complex torus.


title: Dimension of linear systems: a combinatorial and differential approach

abstract:
  We give upper-bounds for the dimension of some linear systems. The theorem
improves the differential Horace method introduced by Alexander-Hirschowitz,
and was conjectured by Simpson. Possible applications are the calculus of the
dimension of linear systems of hypersurfaces in a projective space $\PP^n$ with
generically prescribed singularities, and the calculus of collisions of fat
points in $\PP^2$. These applications will be treated independently but a
simple example in the introduction explains how the theorem will be used.


title:
On fibre space structures of a projective irreducible symplectic
  manifold

abstract:
  In this note, we investigate fibre space structures of a projective
irreducible symplectic manifold. We prove that an 2n-dimensional projective
irreducible symplectic manifold admits only an n-dimensional fibration over a
Fano variety which has only Q-factorial log-terminal singularities and whose
Picard number is one. Moreover we prove that a general fibre is an abelian
variety up to finite unramified cover, especially, a general fibre is an
abelian surface for 4-fold.


title: Skew Schubert functions and the Pieri formula for flag manifolds

abstract:
  We show the equivalence of the Pieri formula for flag manifolds and certain
identities among the structure constants, giving new proofs of both the Pieri
formula and of these identities. A key step is the association of a symmetric
function to a finite poset with labeled Hasse diagram satisfying a symmetry
condition. This gives a unified definition of skew Schur functions, Stanley
symmetric function, and skew Schubert functions (defined here). We also use
algebraic geometry to show the coefficient of a monomial in a Schubert
polynomial counts certain chains in the Bruhat order, obtaining a new
combinatorial construction of Schubert polynomials.


title:
An algebraic cell decomposition of the nonnegative part of a flag
  variety

abstract:
  We study the nonnegative part B_{\ge 0} of the flag variety of a reductive
algebraic group G, as defined by Lusztig. Using positivity properties of the
canonical basis it is shown that B_{\ge 0} has an algebraic cell decomposition
indexed by pairs w\le w' of the Weyl group. This result was conjectired by
Lusztig in [Lu; Progress in Math 123].


title: Convex polytopes and linear algebra

abstract:
  This paper defines, for each convex polytope $\Delta$, a family $H_w\Delta$
of vector spaces. The definition uses a combination of linear algebra and
combinatorics. When what is called exact calculation holds, the dimension
$h_w\Delta$ of $H_w\Delta$ is a linear function of the flag vector $f\Delta$.
It is expected that the $H_w\Delta$ are examples, for toric varieties, of the
new topological invariants introduced by the author in "Local-global
intersection homolog" (preprint alg-geom/9709011).


title: The Gross-Kohnen-Zagier theorem in higher dimensions

abstract:
  The Gross-Kohnen-Zagier theorem describes Heegner points on a modular curve
in terms of coefficients of modular forms. We give another proof of this
theorem which generalizes to higher dimensions.


title: A sparse effective Nullstellensatz

abstract:
  We present bounds for the sparseness and for the degrees of the polynomials
in the Nullstellensatz. Our bounds depend mainly on the unmixed volume of the
input polynomial system. The degree bounds can substantially improve the known
ones when this polynomial system is sparse, and they are, in the worst case,
simply exponential in terms of the number of variables and the maximum degree
of the input polynomials.


title:
Algorithmic Computation of Local Cohomology Modules and the
  Cohomological Dimension of Algebraic Varieties

abstract:
  In this paper we present algorithms that compute certain local cohomology
modules associated to a ring of polynomials containing the rational numbers. In
particular we are able to compute the local cohomological dimension of
algebraic varieties in characteristic zero. Our approach is based on the theory
of D-modules.


title: Ladders on Fano varieties

abstract:
  We prove the existence of Fano ladders on weak log Fano varieties of coindex
less than 4, as an application of adjunction and nonvanishing.


title: Special Lagrangian Fibrations I: Topology

abstract:
  In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric
relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y
are a mirror pair of such manifolds, then X should possess a special Lagrangian
torus fibration $f:X\to B$ such that Y is obtained by dualizing the fibration
f. This leaves a huge amount to be done to verify such conjectures. This paper
takes a speculative point of view, in that it assumes that a special Lagrangian
torus fibration exists on X. We address a number of questions of a topological
nature: what is the relationship between the cohomology of X and the cohomology
of the dual fibration? what kind of information does the Leray spectral
sequence for f contain? what is the relationship between the topological (1,1)
couplings of the dual of f and the (1,n-1)-couplings of X in the large complex
structure limit? These questions are shown to have nice answers if a key
conjecture about the monodromy diffeomorphisms about a large complex structure
limit point holds. Roughly put, this conjecture says that monodromy about a
large complex structure limit point can be described as a very natural
generalization of a Dehn twist for an elliptic curve. Given this conjecture, we
show, among other results, that the large complex radius limit of the (1,n-1)
couplings on X coincide with the topological (1,1) couplings on Y, and if dim
X=3, the Leray filtration and weight filtrations of the mixed Hodge structure
coincide, as conjectured by myself and P.M.H. Wilson, and independently by D.
Morrison.


title: Inequalities for semistable families of arithmetic varieties

abstract:
  In this paper, we will consider a generalization of Bogomolov's inequality
and Cornalba-Harris-Bost's inequality to semistable families of arithmetic
varieties under the idea that geometric semistability implies a certain kind of
arithmetic positivity. The first one is an arithmetic analogue of the relative
Bogomolov's inequality proved by the second author. We also establish the
arithmetic Riemann-Roch formulae for stable curves over regular arithmetic
varieties and generically finite morphisms of arithmetic varieties.


title: On the index of a vector field at an isolated singularity

abstract:
  We consider manifolds with isolated singularities, i.e., topological spaces
which are manifolds (say, $C^\infty$--) outside discrete subsets (sets of
singular points). For (germs of) manifolds with, so called, cone--like
singularities, a notion of the index of an isolated singular point of a vector
field is introduced. There is given a formula for the index of a gradient
vector field on a (real) isolated complete intersection singularity. The
formula is in terms of signatures of certain quadratic forms on the
corresponding spaces of thimbles.


title: On The Projective Normality of Smooth Surfaces of degree nine

abstract:
  We investigate the projective normality of smooth, linearly normal surfaces
of degree 9. All non projectively normal surfaces which are not scrolls over a
curve are classified. Results on the projective normality of surface scrolls
are also given. One of the reasons that brought us to look at this question is
our desire to find examples for a long standing problem in adjunction theory.
Andreatta followed by a generalization by Ein and Lazarsfeld posed the problem
of classifying smooth n-dimensional varieties (X,L) polarized with a very ample
line bundle L, such that the adjoint linear system |H| = |K + (n-1)L| gives an
embedding which is not projectively normal. After a detailed check of the non
projectively normal surfaces found in this work no examples were found except
possibly a blow up of an elliptic P^1-bundle whose existence is uncertain.


title: Isolated rational curves on K3-fibered Calabi-Yau threefolds

abstract:
  We study each of the 16 types of complete intersection Calabi-Yau threefolds
in projective n-space times the projective line, for various n, and prove
existence of isolated rational curves of bidegree (d,0) for all positive
integers d on a general threefold of each of these types. As a corollary we
find isolated rational curves in all degrees on a general CICY of each of the 5
possible types in usual projective n-space.


title: Canonical rational equivalence of intersections of divisors

abstract:
  We consider the operation of intersecting with a locally principal Cartier
divisor (i.e., a Cartier divisor which is principal on some neighborhood of its
support). We describe this operation explicitly on the level of cycles and
rational equivalences and as a corollary obtain a formula for rational
equivalence between intersections of two locally principal Cartier divisors.
Such canonical rational equivalence applies quite naturally to the setting of
algebraic stacks. We present two applications: (i) a simplification of the
development of Fulton-MacPherson-style intersection theory on Deligne-Mumford
stacks, and (ii) invariance of a key rational equivalence under a certain group
action (which is used in developing the theory of virtual fundamental classes
via intrinsic normal cones).


title: Invariant stable bundles over modular curves X(p)

abstract:
  We study stable vector bundles over the modular curve X(p) corresponding to
the principal congruence subgroup of the modular group of prime level p which
are invariant with respect to its automorphism group.


title: Quasi-algebraic geometry of curves I. Riemann-Roch theorem and Jacobian

abstract:
  We discuss an analogue of Riemann-Roch theorem for curves with an infinite
number of handles. We represent such a curve X by its Shottki model, which is
an open subset U of CP^{1} with infinite union of circles as a boundary. An
appropriate bundle on X is \omega^{1/2} \otimes L, L being a bundle with (say)
constants as gluing conditions on the circles. An admissible section of an
appropriate bundle on X is a holomorphic half-form on U with given gluing
conditions and H^{1/2}-smoothness condition.
  We study the restrictions on the mutual position of the circles and the
gluing constants which guarantee the finite dimension of the space of
appropriate sections of admissible bundles, and make the Riemann-Roch theorem
hold. The resulting Jacobian variety is described as an infinite-dimension
analogue of a torus.


title: The product formula for Gromov-Witten invariants

abstract:
  We prove that the system of Gromov-Witten invariants of the product of two
varieties is equal to the tensor product of the systems of Gromov-Witten
invariants of the two factors.


title: Moduli of Trigonal Curves

abstract:
  We study the moduli of trigonal curves. We establish the exact upper bound of
${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any
fibration $X\to B$ of stable curves with smooth general member is the ratio
$\delta_B/\lambda_B$ of the restrictions of the boundary class $\delta$ and the
Hodge class $\lambda$ on the moduli space $\bar{\mathfrak{M}}_g$ to the base
$B$. We associate to a trigonal family $X$ a canonical rank two vector bundle
$V$, and show that for Bogomolov-semistable $V$ the slope satisfies the
stronger inequality ${\delta_B}/{\lambda_B}\leq 7+{6}/{g}$. We further describe
the rational Picard group of the {trigonal} locus $\bar{\mathfrak T}_g$ in the
moduli space $\bar{\mathfrak{M}}_g$ of genus $g$ curves. In the even genus
case, we interpret the above Bogomolov semistability condition in terms of the
so-called Maroni divisor in $\bar{\mathfrak T}_g$.


title: Generalized Reed-Muller codes and curves with many points

abstract:
  The weight hierarchy of generalized Reed-Muller codes over arbitrary finite
fields was determined by Heijnen and Pellikaan. In this paper we produce curves
over finite fields with many points which are closely related to this weight
hierarchy.


title: Un th\'eor\`eme de Rao pour les familles de courbes gauches

abstract:
  The aim of this paper is to prove a generalization of a theorem of Rao for
families of space curves, which caracterizes the biliaison classes of curves.
First we introduce the concept of pseudo-isomorphism of coherent sheaves, which
generalizes the concept of stable isomorphism. An N-type resolution for a
family of curves $C$ over the local ring $A$, defined by an ideal $J$, is an
exact sequence $0\to P\to N\to J\to 0$ where $N$ is a locally free sheaf on
$P^3_A$ and $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. We prove the
two following results, when the residual field of $A$ is infinite : 1. Let $C$
and $C'$ be two flat families of space curves over the local ring $A$. Then $C$
and $C'$ are in the same biliaison class if and only if their ideals $J$ and
$J'$ are pseudo-isomorphic, up to a shift. 2. Let $C$ and $C'$ be two flat
families of space curves over the local ring $A$, with N-type resolutions,
involving sheaves $N$ and $N'$. Then $C$ and $C'$ are in the same biliaison
class if and only if $N$ and $N'$ are pseudo-isomorphic, up to a shift.


title: Generation of $k$-jets on Toric Varieties

abstract:
  In this notes we study $k$-jet ample line bundles $L$ on a non singular toric
variety $X$, i.e. line bundles with global sections having arbitrarily
prescribed $k$-jets at a finite number of points. We introduce the notion of an
associated $k$-convex $\D$-support function, $\psi_L$, requiring that the
polyhedra $P_L$ has edges of length at least $k$. This translates to the
property that the intersection of $L$ with the invariant curves, associated to
every edge, is $\geq k$. We also state an equivalent criterion in terms of a
bound of the Seshadri constant $\e(L,x)$. More precisely we prove the
equivalence of the following: (1) $L$ is $k$-jet ample; (2) $L\cdot C\geq k$,
for any invariant curve $C$; (3) $\psi_L$ is $k$-convex; (4) the Seshadri
constant $\e(L,x)\geq k$ for each $x\in X$.


title: On a conjecture of Lange

abstract:
  Let C be a projective smooth curve of genus g> 1. Let E be a vector bundle of
rank r on C. For each integer r'<r, associate to E the invariant
s_{r'}(E)=r'deg(E)-rdeg(E') where E'is a subbundle of E of rank r' and maximal
degree. For every r', one can stratify the moduli space of stable vector
bundles according to the value of the invariant. Lange's conjecture says that
this strata are non-empty and of the right dimension if s_{r'}>0. The
conjecture has recently been solved thanks to work of Lange- Narasimhan,
Lange-Brambila-Paz, Ballico and the authors. The purpose of this paper is to
give a simpler proof of the result valid without further assumptions. The
method of proof provides additional information on the geometry of the strata.
We can prove that each strata (which is irreducible) is contained in the
closure of the following one. We also show the unicity of the maximal subbundle
when s\le r'(r-r')(g-1). Our methods can be used to study twisted Brill-Noether
loci and to give a new proof of Hirschowitz Theorem about the non-speciality of
the tensor product of generic vector bundles.


title: Birational Calabi--Yau n-folds have equal Betti numbers

abstract:
  Let X and Y be two smooth projective n-dimensional algebraic varieties X and
Y over C with trivial canonical line bundles. We use methods of p-adic analysis
on algebraic varieties over local number fields to prove that if X and Y are
birational, they have the same Betti numbers.


title: Cohomology of complete intersections in toric varieties

abstract:
  This paper explicitly describes Hodge structures of complete intersections of
ample hypersurfaces in compact simplicial toric varieties.


title:
Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete
  Intersections in Grassmannians

abstract:
  In this paper we show that conifold transitions between Calabi-Yau 3-folds
can be used for the construction of mirror manifolds and for the computation of
the instanton numbers of rational curves on complete intersection Calabi-Yau
3-folds in Grassmannians. Using a natural degeneration of Grassmannians
$G(k,n)$ to some Gorenstein toric Fano varieties $P(k,n)$ with conifolds
singularities which was recently described by Sturmfels, we suggest an explicit
mirror construction for Calabi-Yau complete intersections $X \subset G(k,n)$ of
arbitrary dimension. Our mirror construction is consistent with the formula for
the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational
quantum cohomology of Grassmannians.


title: Real deformations and complex topology of plane curve singularities

abstract:
  This is the paper as published. The topology of a complex plane curve
singularity with real branches is deduced from any real deformation having
delta crossings. An example of the computation of the global geometric
monodromy of a polynomial mapping is added.


title:
The Geometry of the Space of Holomorphic Maps from a Riemann Surface
  into Complex Projective Space

abstract:
  In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint
preserving) holomorphic maps of a given degree k from a Riemann surface M of
genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric
products of the surface as well as the geometry of the associated Abel-Jacobi
maps, we construct an explicit topological compactification of Hol(M,P{n},k)
which allow us to construct spectral sequences with known E_1-terms that can be
used to determine the homology groups of Hol(M,P{n},k). Complete calculations
are given for the elliptic case (g=1). Complete rational calculations are also
given for hyperelliptic curves.


title: Minimal even sets of nodes

abstract:
  We extend some results on even sets of nodes which have been proved for
surfaces up to degree 6 to surfaces up to degree 10. In particular, we give a
formula for the minimal cardinality of a nonempty even set of nodes.


title: Hyperkaehler structures on total spaces of holomorphic cotangent bundles

abstract:
  Let $M$ be a Kaehler manifold, and consider the total space $T^*M$ of the
cotangent bundle to $M$. We show that in the formal neighborhood of the zero
section $M \subset T^*M$ the space $T^*M$ admits a canonical hyperkaehler
structure, compatible with the complex and holomorphic symplectic structures on
$T^*M$. The associated hyperkaehler metric $h$ coincides with the given Kaehler
metric on the zero section $M \subset T^*M$. Moreover, $h$ is invariant under
the canonical circle action on $T^*M$ by dilatations along the fibers of $T^*M$
over $M$. We show that a hyperkaehler structure with these properties is
unique. When the Kaehler metric on $M$ is real-analytic, we show that this
formal hyperkaehler structure can be extended to an open neighborhood of the
zero section. We also prove a hyperkaehler analog of the Darboux-Weinstein
Theorem. To prove these results, we use the machinery of $R$-Hodge structures,
following Deligne and Simpson.


title: Injective resolutions of BG and derived moduli spaces of local systems

abstract:
  It was suggested on several occasions by Deligne, Drinfeld and Kontsevich
that all the moduli spaces arising in the classical problems of deformation
theory should be extended to natural "derived" moduli spaces which are always
smooth in an appropriate sense and whose tangent spaces involve the entire
cohomology of the sheaf of infinitesimal automorphisms, not just H^1. In this
note we give an algebraic construction of such an extension for the simplest
class of moduli spaces, namely for moduli of local systems (representations of
the fundamental group).


title: On the infinitesimal rigidity of homogeneous varieties

abstract:
  Let $X\subset P^N$ be a variety (respectively a patch of an analytic
submanifold) and let $x\in X$ be a general point. We show that if the
projective second fundamental form of $X$ at $x$ is isomorphic to the second
fundamental form of a point of a Segre $P^n\times P^m$, $n,m\geq 2$, a
Grassmaniann $G(2,n+2)$, $n\geq 4$, or the Cayley plane $OP^2$, then $X$ is the
corresponding homogeneous variety (resp. a patch of the corresponding
homogeneous variety). If the projective second fundamental form of $X$ at $x$
is isomorphic to the second fundamental form of a point of a Veronese
$v_2(P^n)$ and the Fubini cubic form of $X$ at $x$ is zero, then $X=v_2(P^n)$
(resp. a patch of $v_2(P^n)$). All these results are valid in the real or
complex analytic categories and locally in the $C^{\infty}$ category if one
assumes the hypotheses hold in a neighborhood of any point $x$. As a byproduct,
we show that the systems of quadrics $I_2(P^{m-1}\sqcup P^{n-1}), I_2(P^1\times
P^{n-1})$ and $I_2(S_5)$ are stable in the sense that if $A_t\subset S^2T^*$ is
an analytic family such that for $t\neq 0$, $A_t\simeq A$, then $A_0\simeq A$.
  We also make some observations related to the Fulton-Hansen connectedness
theorem.


title:
Irreducibility of the moduli space of vector bundles on surfaces and
  Brill-Noether theory on singular curves

abstract:
  We prove the irreducibility of the moduli space of rank 2 semistable torsion
free sheaves (with a generic polarization and any value of c_2) on a K3 or a
del Pezzo surface. In the case of a K3 surface, we need to prove a result on
the connectivity of the Brill-Noether locus for singular curves on the surface.
In the case of a del Pezzo surface, we reduce the problem to the case of P^2 by
first relating the moduli spaces of the plane and the blown-up plane, and then
studying how the moduli space changes when we change the polarization.


title: The universal rank-(n-1) bundle on G(1,n) restricted to subvarieties

abstract:
  We classify those smooth (n-1)-folds in G(1,n) for which the restriction of
the rank-(n-1) universal bundle has more than n+1 independent sections. As an
aplication, we classify also those (n-1)-folds for which that bundle splits.


title: Involutions on Moduli Spaces and Refinements of the Verlinde Formula

abstract:
  The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant
over a complex curve carries involutions naturally associated to 2-torsion
points on the Jacobian of the curve. For every lift of a 2-torsion point to a
4-torsion point, we define a lift of the involution to the determinant line
bundle $\L$. We obtain an explicit presentation of the group generated by these
lifts in terms of the order 4 Weil pairing. This is related to the triple
intersections of the components of the fixed point sets in $M$, which we also
determine completely using the order 4 Weil pairing. The lifted involutions act
on the spaces of holomorphic sections of powers of $\L$, whose dimensions are
given by the Verlinde formula. We compute the characters of these vector spaces
as representations of the group generated by our lifts, and we obtain an
explicit isomorphism (as group representations) with the
combinatorial-topological TQFT-vector spaces of [BHMV]. As an application, we
describe a `brick decomposition', with explicit dimension formulas, of the
Verlinde vector spaces. We also obtain similar results in the twisted (i.e.,
degree one) case.


title: Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields

abstract:
  We construct a generalization of the variations of Hodge structures on
Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0
Gromov-Witten invariants


title: On the k-normality of some projective manifolds

abstract:
  A long standing conjecture, known to us as the Eisenbud Goto conjecture,
states that an n-dimensional variety embedded with degree $d$ in the $N$-
dimensional projective space is $(d-(N-n)+1)$-regular in the sense of
Castelnuovo-Mumford.
  In this work the conjecture is proved for all smooth varieties $X$ embedded
by the complete linear system associated with a very ample line bundle $L$ such
that $\Delta (X,L) \le 5$ where $\Delta (X,L) = \dim{X} + \deg{X} -h^0(L).$ As
a by-product of the proof of the above result the projective normality of a
class of surfaces of degree nine in $\Pin{5}$ which was left as an open
question in a previous work of the second author and S. Di Rocco
alg-geom/9710009 is established. The projective normality of scrolls $X
=\Proj{E}$ over a curve of genus 2 embedded by the complete linear system
associated with the tautological line bundle assumed to be very ample is
investigated. Building on the work of Homma and Purnaprajna and Gallego
alg-geom/9511013, criteria for the projective normality of three-dimensional
quadric bundles over elliptic curves are given, improving some results due to
D. Butler.


title: On the variety of rational space curves

abstract:
  We consider the locus of irreducible nonsingular rational curves of degree d
Pn, n>2, meeting a generic collection of linear subspaces. When this locus is 0
(resp 1)- dimensional, we compute (recursively) its degree (resp. geometric
genus). The method is completely elementary and similar to that of
(alg-geom/9704004, alg-geom/9708013), where the case n=2 was considered.


title:
On higher order embeddings of Fano threefolds by the anticanonical
  linear system

abstract:
  The map given by the anticanonical bundle of a Fano manifold is investigated
with respect to a number of natural notions of higher order embeddings of
projective manifolds. This is of importance in the understanding of higher
order embeddings of the special varieties of adjunction theory, which are
usually fibered by special Fano manifolds. An analysis is carried out of the
higher order embeddings of the special varieties of adjunction theory that
arise in the study of the first and second reductions. Special attention is
given to determining the order of the anticanonical embeddings of the three
dimensional Fano manifolds which have been classified by Iskovskih, Mori, and
Mukai and also of the Fano complete intersections in $\pn N$.


title: Resonant Hypergeometric Systems and Mirror Symmetry

abstract:
  The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they
give solutions for certain resonant systems of GKZ hypergeometric differential
equations. For this some complex parameters in the Gamma-series are replaced by
nilpotent elements from a ring $R_{A,T}$. The adapted Gamma-series is a
function $\Psi$ with values in the finite dimensional vector space
$R_{A,T}\otimes C$. Applications of these results in the context of toric
Mirror Symmetry are described. Building on work of Batyrev we show that the
relative cohomology module of a certain hypersurface in a torus is a GKZ
hypergeometric $D$-module which over an appropriate domain is isomorphic to the
trivial $D$-module $R_{A,T}\otimes O_T$, where $O_T$ is the sheaf of
holomorphic functions on this domain. The isomorphism is explicitly given by
adapted Gamma-series. As a result one finds the periods of a holomorphic
differential form of degree $d$ on a $d$-dimensional Calabi-Yau manifold,
needed for the B-model side input to Mirror Symmetry. Relating our work with
that of Batyrev and Borisov we interpret the ring $\cR_{\sA,\gT}$ as the
cohomology ring of a toric variety and a certain principal ideal in it as a
subring of the Chow ring of a Calabi-Yau complete intersection. This
interpretation takes place on the A-model side of Mirror Symmetry.


title:
Pure Hodge structure on the $L_2$-cohomology of varieties with isolated
  singularities

abstract:
  Let $V$ be a complex projective variety with isolated singularities. Let the
smooth part be given the metric induced by a projective imbedding. Then we
develop the $L_2$ harmonic theory and construct a pure Hodge structure on the
$L_2$-cohomology of $V$. If the dimension of $V$ is two, we put a cohomological
Hodge structure on its complex of sheaves of $L_2$ forms.


title: Approximation by smooth curves near the tangent cone

abstract:
  We show that through a point of an affine variety there always exists a
smooth plane curve inside the ambient affine space, which has the multiplicity
of intersection with the variety at least 3.
  This result has an application to the study of affine schemes.


title: Some fundamental problems on real-analytic sets

abstract:
  The category of real-analytic sets and real-analytic maps is the most
important category in application. However, in spite of efforts by F. Bruhat,
H. Cartan, H. Whitney et al., the basic theory of real-analytic category does
not yet seem to be well-developed. In this article I would like to point out
several basic problems.


title: Beyond the Manin obstruction

abstract:
  We construct a (smooth, projective) surface over the field of rational
numbers, which is a counterexample to the Hasse principle not accounted for by
the Manin obstruction. The construction relies on the classical 4-descent on
elliptic curves. By combining the Manin obstruction with descent we define a
refinement of the Manin obstruction which for surfaces of the type considered
suffices to explain the absence of a rational point.


title: Construction de familles minimales de courbes gauches

abstract:
  Let $A$ be a local noetherian ring and $N$ be a locally sheaf on the
projective space $P^3_A$ : one proves easily that there exists a family $C$ of
(smooth connected) curves contained in $P^3_A$, flat over $A$, and an integer
$h$ such that the ideal sheaf $J$ of $C$ has a resolution $0\to P\to N\to J\to
0$ where $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. In this paper
we determine, for a given sheaf $N$, all the families of curves with such a
resolution, especially the minimal ones (corresponding to the minimum value of
$h$). It gives a description of the biliaison class related to $N$, and a tool
for constructing families of space curves.


title:
Stringy Hodge numbers of varieties with Gorenstein canonical
  singularities

abstract:
  We introduce the notion of stringy E-function for an arbitrary normal
irreducible algebraic variety X with at worst log-terminal singularities. We
prove some basic properties of stringy E-functions and compute them explicitly
for arbitrary Q-Gorenstein toric varieties. Using stringy E-functions, we
propose a general method to define stringy Hodge numbers for projective
algebraic varieties with at worst Gorenstein canonical singularities. This
allows us to formulate the topological mirror duality test for arbitrary
Calabi-Yau varieties with canonical singularities. In Appendix we explain
non-Archimedian integrals over spaces of arcs. We need these integrals for the
proof of the main technical statement used in the definition of stringy Hodge
numbers.


title:
Semiinfinite Flags. II. Local and Global Intersection Cohomology of
  Quasimaps' Spaces

abstract:
  For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the
projective line $C$ to the flag variety of $G$. We prove that the global
Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and
compute its generating function. We define an action of the Langlands dual Lie
algebra $g^L$ on this cohomology. We present a new geometric construction of
the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of
$g^L$. It is realized in the Ext-groups of certain perverse sheaves on
Quasimaps' spaces, and it is equipped with a canonical basis numbered by the
irreducible components of certain algebraic cycles, isomorphic to the
intersections of semiinfinite orbits in the affine Grassmannian of $G$. We
compute the stalks of the IC-sheaves on the Schubert strata closures in $Q$.
They carry a natural pure Tate Hodge structure, and their generating functions
are given by the generic affine Kazhdan-Lusztig polynomials. In the Appendix we
prove that Kontsevich's space of stable maps provides a natural resolution of
$Q$.


title:
Constructing curves over finite fields with many points by solving
  linear equations

abstract:
  In this note is we exhibit an elementary method to construct explicitly
curves over finite fields with many points. Despite its elementary character
the method is very efficient and can be regarded as a partial substitute for
the use of class field theory.


title: Hypergeometric functions on reductive groups

abstract:
  The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V.
Zelevinsky and the author, is defined for a set A of characters of an algebraic
torus. In this paper we propose a generalization of the theory where the torus
is replaced by an arbitrary reductive group H and A is a set of irreducible
representations of H. The functions are thus defined on the space M_A of
functions on H spanned by the matrix elements of representations from A. The
properties of the system are related to the geometry of a certain algebraic
variety X_A, which belongs to the class of group compactifications studied by
De Concini and Procesi. We develop the theory of Euler integral representations
for these generalized hypergeometric functions (with integrals taken over
cycles in H). We also construct the analogs of hypergeometric series, by
expanding the delta-function along a subgroup into a power series and taking
the termwise Fourier transform.


title: A conjectural generating function for numbers of curves on surfaces

abstract:
  I give a conjectural generating function for the numbers of $\delta$-nodal
curves in a linear system of dimension $\delta$ on an algebraic surface. It
reproduces the results of Vainsencher for the case $\delta\le 6$ and
Kleiman-Piene for the case $\delta\le 8$. The numbers of curves are expressed
in terms of five universal power series, three of which I give explicitly as
quasimodular forms. This gives in particular the numbers of curves of arbitrary
genus on a K3 surface and an abelian surface in terms of quasimodular forms,
generalizing the formula of Yau-Zaslow for rational curves on K3 surfaces. The
coefficients of the other two power series can be determined by comparing with
the recursive formulas of Caporaso-Harris for the Severi degrees in $\P_2$. We
verify the conjecture for genus 2 curves on an abelian surface. We also discuss
a link of this problem with Hilbert schemes of points.


title:
First quantum correction for the moduli space of stable bundles over a
  Riemann surface

abstract:
  We compute some Gromov-Witten invariants of the moduli space of odd degree
rank two stable vector bundles over a Riemann surface of any genus. Next we
find the first correction term for the quantum product of this moduli space and
hence get the two leading terms of the relations satisfied by the natural
generators of its quantum cohomology. Finally, we use this information to get a
full description of the quantum cohomology of the moduli space when the genus
of the Riemann surface is 3.


title:
Topological Triviality of $\mu$-constant Deformations of Type f(x) +
  tg(x)

abstract:
  We show that every $\mu$-constant family of isolated hypersurface
singularities of type f(x) + tg(x), where t is a parameter, is topologically
trivial. In the proof we construct explicitely a vector field trivializing the
family. The proof uses only the curve selection lemma and hence, for an
appropriately translated statement, also works over the reals. Some
applications to the study of singularities at infinity of complex polynomials
are given.


title:
Weak approximation, Brauer and R-equivalence in algebraic groups over
  arithmetical fields

abstract:
  We prove some new relations between weak approximation and some rational
equivalence relations (Brauer and R-equivalence) in algebraic groups over
arithmetical fields. By using weak approximation and local - global approach,
we compute completely the group of Brauer equivalence classes of connected
linear algebraic groups over number fields, and also completely compute the
group of R-equivalence classes of connected linear algebraic groups $G$, which
either are defined over a totally imaginary number field, or contains no
anisotropic almost simple factors of exceptional type $^{3,6}\D_4$, nor $\E_6$.
We discuss some consequences derived from these, e.g., by giving some new
criteria for weak approximation in algebraic groups over number fields, by
indicating a new way to give examples of non stably rational algebraic groups
over local fields and application to norm principle.


title: Boundary Manifolds of Line Arrangements

abstract:
  In this paper we describe the complement of real line arrangements in the
complex plane in terms of the boundary three-manifold of the line arrangement.
We show that the boundary manifold of any line arrangement is a graph manifold
with Seifert fibered vertex manifolds, and depends only on the incidence graph
of the arrangement. When the line arrangement is defined over the real numbers,
we show that the homotopy type of the complement is determined by the incidence
graph together with orderings on the edges emanating from each vertex.


title: On the theta divisor of SU(2,1)

abstract:
  Let SU(2,1) be the moduli space of stable rank two vector bundles having
fixed determinant of odd degree over a compact Riemann surface C. In this paper
it is shown that the Theta divisor of SU(2,1) is very ample for every C. The
proof is related to the study of the base locus of the pencil of divisors
2-theta in the Jacobian of C which is naturally associated to a point of
SU(2,1).


title: Chern Classes of Bundles over Rational Surfaces

abstract:
  Consider the blow up $\pi: \widetilde{X} \to X$ of a rational surface $X$ at
a point. Let $\widetilde{V}$ be a holomorphic bundle over $\widetilde{X}$ whose
restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$
and define $V =(\pi_*\widetilde{V})^{\vee \vee}.$ Friedman and Morgan gave the
following bounds for the second Chern classes $j \leq c_2(\widetilde{V}) -
c_2(V) \leq j^2.$ We show that these bounds are sharp.


title: Stringy Hodge numbers and Virasoro algebra

abstract:
  Let $X$ be an arbitrary smooth $n$-dimensional projective variety. It was
discovered by Libgober and Wood that the product of the Chern classes
$c_1(X)c_{n-1}(X)$ depends only on the Hodge numbers of $X$. This result has
been used by Eguchi, Jinzenji and Xiong in their approach to the quantum
cohomology of $X$ via a representation of the Virasoro algebra with the central
charge $c_n(X)$. In this paper we define for singular varieties $X$ a rational
number $c_{st}^{1,n-1}(X)$ which is a stringy version of the number
$c_1c_{n-1}$ for smooth $n$-folds. We show that the number $c_{st}^{1,n-1}(X)$
can be expressed in the same way using the stringy Hodge numbers of $X$. Our
results provides an evidence for the existence of an approach to quantum
cohomology of singular varieties $X$ via a representation of the Virasoro
algebra whose central charge is the rational number $e_{st}(X)$ which equals
the stringy Euler number of $X$.


title:
Semistable reduction in characteristic 0 for families of surfaces and
  three-folds

abstract:
  Let F:X->B be a morphism of varieties in characteristic zero. The problem of
semistable reduction of F was stated as a problem in the combinatorics of
polyhedral complexes by Abramovich and Karu (alg-geom/9707012). In this paper
we solve the combinatorial problem when the relative dimension of F is not
bigger than 3.


title: A geometric approach to the fundamental lemma for unitary groups

abstract:
  We consider from a geometric point of view the conjectural fundamental lemma
of Langlands and Shelstad for unitary groups over a local field of positive
characteristic. We introduce projective algebraic varieties over the finite
residue field $k$ and interpret the conjecture in this case as a remarkable
identity between the number of $k$-rational points of them. We prove the
corresponding identity for the numbers of $k_f$-rational points, for any
extension of even degree $f$ of $k$. The proof uses the local intersection
theory on a regular surface and Deligne's theory of intersection multiplicities
with weights. We also discuss a possible descent argument that uses $\ell$-adic
cohomology to treat extensions of odd degree as well.


title: Equations of the moduli of pointed curves in the infinite Grassmannian

abstract:
  The main result of this paper is the explicit computation of the equations
defining the moduli space of triples $(C,p,z)$ (where $C$ is an integral and
complete algebraic curve, $p$ a smooth rational point and $z$ a formal
trivialization around $p$) in the infinite Grassmannian of $k((t))$. This is
achieved by introducing infinite Grassmannians, tau and Baker-Ahkiezer
functions algebraically and by proving an Addition Formula for tau functions.


title: Cellular Resolutions of Monomial Modules

abstract:
  We construct a canonical free resolution for arbitrary monomial modules and
lattice ideals. This includes monomial ideals and defining ideals of toric
varieties, and it generalizes our joint results with Irena Peeva for generic
ideals.


title: Complements on surfaces

abstract:
  The main result of the paper is a boundedness for $n$-complements on
algebraic surfaces. In addition, applications of this theorem to a
classification of log Del Pezzo surfaces and of birational contractions for
3-folds are formulated.


title: Cycles on Siegel 3-folds and derivatives of Eisenstein series

abstract:
  We consider the Siegel modular variety of genus 2 and a p-integral model of
it for a good prime p>2, which parametrizes principally polarized abelian
varieties of dimension two with a level structure. We consider cycles on this
model which are characterized by the existence of certain special
endomorphisms, and their intersections. We characterize that part of the
intersection which consists of isolated points in characteristic p only.
Furthermore, we relate the (naive) intersection multiplicities of the cycles at
isolated points to special values of derivatives of certain Eisenstein series
on the metaplectic group in 8 variables.


title: Symmetric matrices and quantum codes

abstract:
  This paper has been withdrawn since a Gilbert-Varshamov bound for general
quantum codes has already appeared in Ekert and Macchiavello, Prys. Rev. Lett.
77, p. 2585, and a Gilbert-Varshamov bound for stabilizer codes connected with
orthogonal geometry, or equivalently, with symmetric matrices as in this paper,
has been proved by Calredbank, Rains, Shor and Sloane, Phys. Rev. Lett. 78, p.
405. I would like to thank Robert Calderbank for pointing out these references
to me.


title: Mathematical Instantons In Characteristic Two

abstract:
  On P3, we show that mathematical instantons in characteristic two are
unobstructed. We produce upper bounds for the dimension of the moduli space of
stable rank two bundles on P3 in characteristic two. In cases where there is a
phenomenon of good reduction modulo two, these give similar results in
characteristic zero. We also give an example of a non-reduced component of the
moduli space in characteristic two.


title: On moduli spaces of 4- or 5-instanton bundles

abstract:
  We study the scheme of multi-jumping lines of an $n$-instanton bundle mainly
for $n\leq 5$. We apply it to prove the irreducibility and smoothness of the
moduli space of 5-instanton. Some particular situations with higher $c_2$ are
also studied.


title: Classification of Exceptional Complements: Elliptic Curve Case

abstract:
  We classify the log del Pezzo surface (S,B) of rank 1 with no 1-,2-,3-,4-, or
6-complements with the additional condition that B has one irreducible
component C which is an elliptic curve, and that C has the coefficient b in B
with (1/n)floor((n+1)b)=1 for n=1,2,3,4, and 6.


title:
Quantum cohomology of the moduli space of stable bundles over a Riemann
  surface

abstract:
  We determine the quantum cohomology of the moduli space of odd degree rank
two stable vector bundles over a Riemann surface $\Sigma$ of any genus. This
work together with dg-ga/9710029 prove that this quantum cohomology is
isomorphic to the instanton Floer cohomology of the three manifold $\Sigma
\times S^1$. (Note: There is some overlap with the previous paper:
alg-geom/9711013).


title: The enumerative geometry of K3 surfaces and modular forms

abstract:
  We prove the conjectures of Yau-Zaslow and Gottsche concerning the number
curves on K3 surfaces. Specifically, let X be a K3 surface and C be a
holomorphic curve in X representing a primitive homology class. We count the
number of curves of geometric genus g with n nodes passing through g generic
points in X in the linear system |C| for any g and n satisfying C^2=2g+2n-2.
  When g=0, this coincides with the enumerative problem studied by Yau and
Zaslow who obtained a conjectural generating function for the numbers.
Recently, Gottsche has generalized their conjecture to arbitrary g in terms of
quasi-modular forms. We prove these formulas using Gromov-Witten invariants for
families, a degeneration argument, and an obstruction bundle computation. Our
methods also apply to P^2 blown up at 9 points where we show that the ordinary
Gromov-Witten invariants of genus g constrained to g points are also given in
terms of quasi-modular forms.


title:
On the classification of hyperbolic root systems of the rank three. Part
  I

abstract:
  It was recently understood that from the point of view of automorphic
Lorentzian Kac-Moody algebras and some aspects of Mirror Symmetry, interesting
hyperbolic root systems should have restricted arithmetic type and a
generalized lattice Weyl vector. One can consider hyperbolic root systems with
these properties as an appropriate hyperbolic analog of the classical finite
and affine root systems.
  This series of papers is devoted to classification of hyperbolic root systems
of restricted arithmetic type and with a generalized lattice Weyl vector
$\rho$, having the rank 3 (it is the first non-trivial rank). In the Part I we
announce classification of the maximal hyperbolic root systems of elliptic
(i.e. $\rho^2 >0$) and parabolic (i.e. $\rho^2=0$) type, having the rank 3. We
give sketch of the proof. Details of the proof and further results (non-maximal
cases) will be given in Part II. Classification for hyperbolic type (i.e.
$\rho^2<0$) and applications will be considered in Part III.


title: Uniform Zariski's Theorem On Fundamental Groups

abstract:
  The Zariski theorem says that for every hypersurface in a complex projective
(resp. affine) space of dimension at least 3 and for every generic plane in the
projective (resp. affine) space the natural embedding generates an isomorphism
of the fundamental groups of the complements to the hypersurface in the plane
and in the space. If a family of hypersurfaces depends algebraically on
parameters then it is not true in general that there exists a plane such that
the natural embedding generates isomorphisms of the fundamental groups of the
complements to each hypersurface from this family in the plane and in the
space. But we show that in the affine case such plane exists after a polynomial
coordinate substitution.


title: Minimal discrepancies of hypersurface singularities

abstract:
  We give an upper bound for the minimal discrepancies of hypersurface
singularities. As an application, we show that Shokurov's conjecture is true
for log-terminal threefolds.


title: On the Euler Characteristic of Generalized Kummer Varieties

abstract:
  We apply the Yau-Zaslow-Beauville method to compute the Euler characteristic
of the generalized Kummer varieties attached to a complex abelian surface (a
calculation also done by Goettsche and Soergel by different methods). It is
related to the number of genus 2 curves on an abelian surface.


title:
Coupled Contact Systems and Rigidity of Maximal Dimensional Variations
  of Hodge Structure

abstract:
  In this article we prove that locally Griffiths' horizontal distribution on
the period domain is given by a generalized version of the familiar contact
differential system. As a consequence of this description we obtain strong
local rigidity properties of maximal dimensional variations of Hodge structure.
For example, we prove that if the weight is odd then there is a unique germ of
maximal dimensional variation of Hodge stucture though every point of the
period domain. Similar results hold if the weight is even with the exception of
one case.