title: Adjoints of ideals in regular local rings

abstract:
  The adjoint of an ideal I in a regular local ring R is the R-ideal
adj(I):=H^0(Y, I\omega_Y), where f:Y -> Spec(R) is a proper birational map with
Y nonsingular and IO_Y invertible, and \omega_f is a canonical relative
dualizing sheaf. (Such an f is supposed to exist.) The basic conjecture is that
I.adj(I^n)=adj(I^{n+1}) whenever n is >= the analytic spread of I. This is a
strong version of the Briancon-Skoda theorem, implying all other known
versions. It follows from the (conjectural) vanishing of H^i(Y,\omega_Y) for
all i>0. When R is essentially of finite type over a char. 0 field, that
vanishing has been deduced by Cutkosky from Kodaira vanishing. It also holds
whenever dim.R = 2; and in this case we can say considerably more about
adjoints, tying in e.g., with classical material on adjoint curves and
conductors.


title:
Algebraic Surfaces with Log-Terminal Singularities and nef Anticanonical
  Class and Reflection Groups in Lobachevsky Spaces. I (Basics of the Diagram
  Method)

abstract:
  In our preprint: "Algebraic surfaces with log-terminal singularities and nef
anticanonical class and reflection groups in Lobachevsky spaces", Preprint
Max-Planck-Institut f\"ur Mathematik, Bonn, (1989) MPI/89-28 (Russian), we had
extended the diagram method to algebraic surfaces with nef anticanonical class
(before, it was applied to Del Pezzo surfaces). The main problem here is that
Mori polyhedron may not be finite polyhedral in this case. In the \S 4 of this
preprint we had shown that this method does work for algebraic surfaces with
nef anticanonical class and log terminal singularities.
  Recently, using in particular these our results, Valery Alexeev (Boundedness
and K^2 for log surfaces, Preprint (1994), alg-geom/942007), obtained strong
and important results for surfaces of general type with semi-log canonical
singularities. The reason is that Del Pezzo surfaces (the diagram method is
especially effective for them) and surfaces of general type "are connected"
through surfaces with numerically zero canonical class. This was the main
reason that we decided to make this English translation of the first part (\S
1--\S 3) of our preprint above with some extensions (where we give omitted
proofs) and minor corrections.
  This part contains basics of the diagram method for generalized reflection
groups in Lobachevsky spaces and for surfaces with nef anticanonical class.


title: Local structure of the moduli space of vector bundles over curves

abstract:
  We analyze the local structure of the moduli space of semi-stable bundles on
a curve. In particular, a complete description of the local structure is given
in the rank 2 case. We obtain as a corollary of this analysis new results about
the Kummer variety and the Coble quartic associated to a canonical genus 3
curve.


title: Deformation theory and Lie algebra homology

abstract:
  A description of a ring of functions on the base of a universal formal
deformation for several moduli problems is given. The answer is given in terms
of a homology group of a certain dg Lie algebra canonically (up to an
essentially unique quasi-isomorphism) associated with a problem.


title: On Riemann-Roch Formulas for Multiplicities

abstract:
  A Theorem due to Guillemin and Sternberg about geometric quantization of
Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says
that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch
number of the symplectically reduced space. Combined with the shifting-trick,
this gives explicit formulas for the multiplicities of the various irreducible
components. One of the assumptions of the Theorem is that the reduction is
regular, so that the reduced space is a smooth symplectic manifold. In this
paper, we prove a generalization of this result to the case where the reduced
space may have orbifold singularities. Our proof uses localization techniques
from equivariant cohomology, and relies in particular on recent work of
Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments
involved, the result also extends to non Kaehlerian settings.


title: Gorenstein Quotient Singularities of Monomial Type in Dimension Three

abstract:
  The purpose of this paper is to construct a crepant resolution of quotient
singularities by finite subgroups of SL(3,C) of monomial type, and prove that
the Euler number of the resolution is equal to the number of conjugacy classes.
This result is a part of conjecture II in previous paper "Crepant resolution of
trihedral singularities" (alg-geom 9404008). These singularities are different
from trihedral, but main idea of the proof is based on the method of trihedral
case.


title: Spin canonical invariants of 4-manifolds and algebraic surfaces

abstract:
  The paper is a colloquial-style discussion of invariants of algebraic
surfaces analogous to the Donaldson polynomials, arising from moduli spaces of
``jumping'' Yang--Mills instantons, or moduli spaces of jumping vector bundles.
The invariants have the following applications: (1) to the Van de Ven
conjecture that the Kodaira dimension is a diffeomorphism invariant; (2) to
proving that algebraic surfaces with $p_g > 0$ have a proper sublattice of
$H^2(X,\Z)$ invariant under diffeomorphism; (3) to proving the same result as
(2) for surfaces with $p_g = 0$, in particular the Barlow surface.


title:
Universal Hodge Bundle and Mumford Isomorphisms on the Abelian Inductive
  Limit of Teichm\"uller Spaces

abstract:
  Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$.
Mumford had proved that, for each fixed genus $g$, there are isomorphisms
asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed
(genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we
obtain a coherent, genus-independent description of the Mumford isomorphisms
over certain infinite-dimensional ``universal'' parameter spaces of compact
Riemann surfaces (with or without marked points). We work with an inductive
limit of Teichm\"uller spaces comprising complex structures on a certain
``solenoidal Riemann surface'', $H_{\infty,abel}$. The main result constructs
these universal $DET$ and Hodge line bundles on this direct limit, equips them
with their Quillen metrics, and proves that they are related by the appropriate
Mumford isomorphisms. Our work can be viewed as a contribution towards a
non-perturbative formulation of the bosonic string theory, since it provides an
interpretation of the Polyakov measure in a genus-independent fashion.


title: Log Smooth Deformation Theory

abstract:
  This paper gives a foundation of log smooth deformation theory. We study the
infinitesimal liftings of log smooth morphisms and show that the log smooth
deformation functor has a representable hull. This deformation theory gives,
for example, the following two types of deformations: (1) relative deformations
of a certain kind of a pair of an algebraic variety and a divisor of it, and
(2) global smoothings of normal crossing varieties. The former is a
generalization of the relative deformation theory introduced by Makio, and the
latter coincides with the logarithmic deformation theory introduced by Kawamata
and Namikawa.


title: Pfaffian Subschemes

abstract:
  A subscheme $X\subset \Bbb P^{n+3}$ of codimension $3$ is {\em Pfaffian} if
it is the degeneracy locus of a skew-symmetric map $f:\cal{E}\spcheck(-t) @>>>
\cal{E}$ with $\cal{E}$ a locally free sheaf of odd rank on $\Bbb P^{n+3}$. It
is shown that a codimension $3$ subscheme $X\subset\Bbb P^{n+3}$ is Pfaffian if
and only if it is locally Gorenstein, subcanonical (i.e.\ $\omega_X\cong\cal
O_X(l)$ for some integer $l$), and the following parity condition holds: if
$n\equiv 0\pmod{4}$ and $l$ is even, then $\chi (\cal O_X (l/2))$ is also even.
The paper includes a modern version of the Horrocks correspondence, stated in
the language of derived categories. A local analogue of the main theorem is
also proved.


title: Infinite-dimensional Lie algebras and the period map for curves

abstract:
  We compute higher-order differentials of the period map for curves and show
how they factor through the corresponding higher Kodaira-Spencer classes. Our
approach is based on the infinitesimal equivariance of the period map, due to
Arbarello and De Concini \cite{AD}.


title:
Some glueing formulas for spin polynomials and a proof of the Van de Ven
  conjecture

abstract:
  We deduce some formulas for the spin invariants of the connected sum of an
arbitrary 4-manifold X, b^+ _2(X) > 0 $ with $\overline {CP^2}$ in terms of the
spin invariants of $X$ and apply this to prove the Van de Ven conjecture.


title:
Combinatorial and algebro-geometric cohomology classes on the moduli
  spaces of curves

abstract:
  Based on the combinatorial description of the moduli spaces of curves
provided by Strebel differentials, Witten and Kontsevich have introduced
combinatorial cohomology classes $W_{(m_0,m_1,m_2,\dots),n}$, and conjectured
that these can be expressed in terms of Mumford-Morita-Miller classes. It is
argued that this link should be provided by a theorem of Di Francesco, Itzykson
and Zuber which relates the derivatives of the Witten-Kontsevich partition
function with respect to one set of variables to the derivatives with respect
to the other set of variables. Two things are shown. First of all that this
works in complex codimension 1. Secondly that in all the cases when it has been
possible to make the Di Francesco, Itzykson and Zuber correpondence explicit
this translates into identities of the type $$
\int_{W_{(m_0,m_1,m_2,\dots),n}}\prod\psi_i^{d_i}
=\int_{\overline{\cal{M}}_{g,n}} X_{(m_0,m_1,m_2,\dots),n}\prod\psi_i^{d_i} $$
where the $X_{(m_0,m_1,m_2,\dots),n}$ are explicit polynomials in the
algebro-geometric classes and the $\psi_i$ are the Chern classes of the point
bundles, for any choice of $d_1,\dots,d_n$.


title: Quadrature Mirror Filters and Loop Groups

abstract:
  In this paper we want to show, that the finite impulse response quadratic
mirror filters (QMF) associated to a tower of grids $\Gamma\subset H=\bf Z^n$
can be identified with a right coset of the subgroup
Fix$(T_{\Gamma^{\perp}}$,Map$ ({\bf T}^n\to U(N)$: poly) of the group of
polynomial loops Map$({\bf T}^n\to U(N)$: poly) with $N=|H/\Gamma|$. The QMF
with some vanishing moments can be identified with cosets of subgroups. The
problem to parameterize all finite impulse response QMF in arbitrary space
dimensions is now equivalent to factorize all polynomial loops.


title: Differential-geometric characterizations of complete intersections

abstract: The exposition has been significantly altered, hopefully improved.


title:
Canonical Generators for the Cohomology of Moduli of Parabolic Bundles
  on Curves

abstract:
  We determine generators of the rational cohomology algebras of moduli spaces
of parabolic vector bundles on a curve, under some `primality' conditions on
the parabolic datum. These generators are canonical in a precise sense. Our
results are new even for usual vector bundles (i.e., vector bundles without
parabolic structure) whose rank is greater than 2 and is coprime to the degree;
in this case, they are generalizations of a theorem of Newstead on the moduli
of vector bundles of rank 2 and odd degree.


title: On the stability of the tangent bundle of Fano manifolds

abstract:
  By using the classification of Fano 3-folds we prove: Let $X$ be Fano 3-fold.
Assume that the tangent bundle $T_X$ of $X$ is not stable (i.e. semi-stable or
unstable). Then $b_2\geq 2$ and a relative tangent sheaf $T_{X/Y}$ of a
contraction $f:X\longrightarrow Y$ of an extremal face on $X$ is a
destabilising subsheaf of $T_X$.


title: Generating Functions in Algebraic Geometry and Sums over Trees

abstract:
  We calculate generating functions for the Poincare polynomials of moduli
spaces of pointed curves of genus zero and of Configuration Spaces of Fulton
and MacPherson. We also prove that contributions of multiple coverings of
curves in a Calabi-Yau threefold calculated via Kontsevich's compactification
coincide with the prediction of Candelas et al., and Aspinwall and Morrison.


title: Reduction of the Manin map modulo $p$

abstract:
  For an abelian variety $A$ over a function field $K$ of characteristic zero,
Manin defined a remarkable additive map $A(K) \ra V$, where $V$ is a vector
space over $K$. We define an analogue of this map in the case of function
fields of characteristic $p$. We then prove that the reduction modulo $p$ of
the Manin map in characteristic zero is the derivative of the Manin map in
characteristic $p$ and that the kernel of the Manin map in characteristic $p$
is the group of points divisible by $p$.


title: Beyond the K\"ahler cone

abstract:
  The moduli space of nonlinear $\sigma$-models on a Calabi--Yau manifold
contains a complexification of the K\"ahler cone of the manifold. We describe a
physically natural analytic continuation process which links the complexified
K\"ahler cones of birationally equivalent Calabi--Yau manifolds. The enlarged
moduli space includes a complexification of Kawamata's ``movable cone''. We
formulate a natural conjecture about the action of the birational automorphism
group on this cone. (Based in part on joint work with Paul Aspinwall and Brian
Greene; submitted to the proceedings of ``Hirzebruch's 65th birthday workshop
in algebraic geometry''.)


title: Natural transformations from constructible functions to homology

abstract:
  For complex projective varieties, all natural transformations from
constructible functions to homology (modulo torsion) are linear combinations of
the MacPherson-Schwartz-Chern classes. (The authors are willing to mail hard
copies of the paper.)


title: Mordell-Weil problem for cubic surfaces

abstract:
  Drawing the secant through two rational points of a cubic surface we can get
the third one. Is the set of rational points finitely generated? We discuss
some numerical data and prove a finite generation statement with respect to a
modified composition law.


title: Real Grassmann Polylogarithms and Chern Classes

abstract:
  In this paper we define real grassmann polylogarithms, which are real single
valued analogues of the grassmann polylogarithms (or higher logarithms) defined
by Hain and MacPherson. We prove the existence of all such real grassmann
polylogs, at least generically. We also prove that the canonical choice of such
an m-polylogarithm represents the Beilinson Chern class on the rank m part of
the algebraic K-theory of the generic point of every complex algebraic variety.
One part of each such grassmann m-polylogarithm is a real, single-valued
function defined generically on the grassmannian of m planes in C^{2m}. We
prove that this function represents the Borel regulator (up to a factor of 2)
on K_{2m-1} of all number fields.


title: Complex monodromy and the topology of real algebraic sets

abstract:
  We study the Euler characteristic of the real Milnor fibres of a real
analytic map, using a relation between complex monodromy and complex
conjugation. We deduce the result of Coste and Kurdyka that the Euler
characteristic of the link of an irreducible algebraic subset of a real
algebraic set is generically constant modulo 4. We generalize this result to
iterated links of ordered families of algebraic subsets.


title:
On certain N--sheeted coverings and numerical semigroups which cannot be
  realized as Weierstrass semigroups

abstract:
  A curve $X$ is said to be of type $(N,\gamma)$ if it is an $N$--sheeted
covering of a curve of genus $\gamma$ with at least one totally ramified point.
A numerical semigroup $H$ is said to be of type $(N,\gamma)$ if it has $\gamma$
positive multiples of $N$ in $[N,2N\gamma]$ such that its $\gamma^{th}$ element
is $2N\gamma $ and $(2\gamma+1)N \in H$. If the genus of $X$ is large enough
and $N$ is prime, $X$ is of type $(N,\gamma)$ if and only if there is a point
$P \in X$ such that the Weierstrass semigroup at $P$ is of type $(N,\gamma)$
(this generalizes the case of double coverings of curves). Using the proof of
this result and the Buchweitz's semigroup, we can construct numerical
semigroups that cannot be realized as Weierstrass semigroups although they
might satisfy Buchweitz's criterion.


title: Hodge numbers of moduli spaces of stable bundles on K3 surfaces

abstract:
  We show that the Hodge numbers of the moduli space of stable rank two sheaves
with primitive determinant on a K3 surface coincide with the Hodge numbers of
an appropriate Hilbert scheme of points on the K3 surface. The precise result
is: Theorem: Let $X$ be a K3 surface, $L$ a primitive big and nef line bundle
and $H$ a generic polarization. If the moduli space of rank two $H$ semi-stable
torsion-free sheaves with determiant $L$ and second Chern class $c_2$ has at
least dimension 10 then its Hodge numbers coincide with those of the Hilbert
scheme of $l:=2c_2-\frac{L^2}{2}-3$ points on $X$.


title: ${\cal H}$-cohomologies versus algebraic cycles

abstract:
  Global intersection theories for smooth algebraic varieties via products in
{\it appropriate}\, Poincar\'e duality theories are obtained. We assume given a
(twisted) cohomology theory $H^*$ having a cup product structure and we let
consider the ${\cal H}$-cohomology functor $X\leadsto H^{\#}_{Zar}(X,{\cal
H}^*)$ where ${\cal H}^*$ is the Zariski sheaf associated to $H^*$. We show
that the ${\cal H}$-cohomology rings generalize the classical ``intersection
rings'' obtained via rational or algebraic equivalences. Several basic
properties e.g.\, Gysin maps, projection formula and projective bundle
decomposition, of ${\cal H}$-cohomology are obtained. We therefore obtain, for
$X$ smooth, Chern classes $c_{p,i} : K_i(X) \to H^{p-i}(X,{\cal H}^p)$ from the
Quillen $K$-theory to ${\cal H}$-cohomologies according with Gillet and
Grothendieck. We finally obtain the ``blow-up formula'' $$H^p(X',{\cal H}^q)
\cong H^p(X,{\cal H}^q)\oplus \bigoplus_{i=0}^{c-2} H^{p-1-i}(Z,{\cal
H}^{q-1-i})$$ where $X'$ is the blow-up of $X$ smooth, along a closed smooth
subset $Z$ of pure codimension $c$. Singular cohomology of associated analityc
space, \'etale cohomology, de Rham and Deligne-Beilinson cohomologies are
examples for this setting.


title: Local Positivity of Ample Line Bundles

abstract:
  Let $L$ be a nef line bundle on a smooth complex projective variety $X$ of
dimension $n$. Demailly has introduced a very interesting invariant --- the
Seshadri constant $\epsilon(L,x)$ --- which in effect measures how positive $L$
is locally near a given point $x \in X$. For instance, Seshadri's criterion for
ampleness may be phrased as stating that $L$ is ample if and only if there
exists a positive number $e > 0$ such that $\epsilon(L,x) > e$ for all $x \in
X$, and if $L$ is VERY ample, then $\epsilon(L,x) \ge 1$ for every $x$. We
prove the somewhat surprising result that in each dimension $n$ there is a
uniform lower bound on the Seshadri constant of an ample line bundle $L$ at a
very general point of $X$. Specifically, $\epsilon(L,x) \ge (1/n) $ for all $x
\in X$ outside the union of countably many proper subvarieties of $X$. Examples
of Miranda show that there cannot exist a bound (independent of $X$ and $L$)
that holds at every point. The proof draws inspiration from two sources: first,
the arguments used to prove boundedness of Fano manifolds of Picard number one;
and secondly some of the geometric ideas involving zero-estimates appearing in
the work of Faltings and others on Diophantine approximation and transcendence
theory. We give some elementary applications of the main theorem to adjoint and
pluricanonical linear series.


title: Cubics, Integrable Systems, and Calabi-Yau Threefolds

abstract:
  In this work we construct an analytically completely integrable Hamiltonian
system which is canonically associated to any family of Calabi-Yau threefolds.
The base of this system is a moduli space of gauged Calabi-Yaus in the family,
and the fibers are Deligne cohomology groups (or intermediate Jacobians) of the
threefolds. This system has several interesting properties: the multivalued
sections obtained as Abel-Jacobi images, or ``normal functions'', of a family
of curves on the generic variety of the family, are always Lagrangian; the
natural affine coordinates on the base, which are used in the mirror
correspondence, arise as action variables for the integrable system; and the
Yukawa cubic, expressing the infinitesimal variation of Hodge structure in the
family, is essentially equivalent to the symplectic structure on the total
space.


title: Moduli of Brill-Noether pairs on algebraic curves

abstract:
  We construct coarse moduli spaces for `Brill-Noether pairs'. Such a pair
consists of a torsion-free sheaf $E$ over an algebraic curve $X$ and a vector
subspace $\Lambda$ of its space of sections $H^0(E)$. The construction works
for an arbitrary singular curve $X$ of pure dimension and for all values of the
positive rational parameter $\alpha$ occurring in the stability condition.
(Hard copies available on request.)


title: On Landsberg's criterion for complete intersections

abstract:
  In his preprint ``Differential-Geometric Characterizations of Complete
Intersections'' (alg-geom/9407002), J.M.Landsberg introduces an elementary
characterization of complete intersections. The proof of this criterion uses
the method of moving frames. The aim of this note is to present an elementary
proof of Landsberg's criterion that is valid over any ground field.


title: A surface of general type with \( p_g =q =0, K^2 =1 \)

abstract:
  We construct a surface of general type with invariants \( \chi = K^2 = 1 \)
and torsion group \( \Bbb{Z}/{2} \). We use a double plane construction by
finding a plane curve with certain singularities, resolving these, and taking
the double cover branched along the resulting smooth curve.


title: Various Notions of Associated Prime Ideals

abstract:
  Three notions of associated prime ideals, which are equivalent in the
noetherian case but differ in the non notherian case, are discussed. Examples
illustrate the scope of the notions.


title: Ternary quartics and 3-dimensional commutative algebras

abstract:
  We find the connection between 3-dimensional commutative algebras with a
trivial trace and plane quartics and its bitangents.


title: Rationality of the Moduli Variety of Curves of Genus 3

abstract: We prove rationality of the moduli variety of curves of genus 3.


title: Lectures on Linear Series

abstract:
  The past decade has witnessed two important new developments in the study of
linear series on algebraic varieties. First, vector bundles have emerged as
powerful tools for analyzing linear series on curves and surfaces. More
recently, the viewpoints and techniques of higher dimensional geometry have
started to play a central role. The purpose of these lectures is to provide a
down-to-earth introduction to some of these new ideas and methods.
  The present notes, written with the assistance of G. Fernandez del Busto, are
a considerably expanded version of a course given at the 1993 Regional Geometry
Institute on Higher Dimensional Complex Geometry at Park City, Utah. While I
hope they may be of interest to any geometers wishing to learn about recent
work on linear series, I have particularly attempted to aim the discusion at a
novice audience. My hope is that with a little faith and effort, these notes
should be accessible to anyone having finished the standard introductory texts
in algebraic geometry. The underlying theme of the lectures is the search for
higher dimensional generalizations of the most basic facts about linear series
on curves. But in an attempt to keep things elementary we work more or less
exclusively on surfaces. Thus we end up discussing one central result -- viz.
Reider's theorem -- from many points of view. Sticking to the case of surfaces
allows one to eliminate many technical complexities, and I hope that parts of
these lectures might therefore serve as a useful first introduction to the
powerful cohomological techniques (involving $Q$-divisors) of contemporary


title:
The Number of Rational Quartics on Calabi-Yau hypersurfaces in Weighted
  Projective Space P(2,1^4)

abstract:
  We compute the number of rational quartics on a general Calabi-Yau
hypersurface in weighted projective space P(2,1^4). The result agrees with the
prediction made by mirror symmetry.


title: Dynkin Graphs and Triangle Singularities

abstract:
  We treat nine of fourteen triangle singularities in Arnold's classification
list of singularities. We consider what kind of combinations of rational double
points can appear on their small deformation fibers. We show their combinations
are described by a simple priciple using Dynkin graphs.


title: Variation of the Gieseker and Uhlenbeck Compactifications

abstract:
  In this article, we study the variation of the Gieseker and Uhlenbeck
compactifications of the moduli spaces of Mumford-Takemoto stable vector
bundles of rank 2 by changing polarizations. Some {\it canonical} rational
morphisms among the Gieseker compactifications are proved to exist and their
fibers are studied. As a consequence of studying the morphisms from the
Gieseker compactifications to the Uhlebeck compactifications, we show that
there is an everywhere-defined {\it canonical} algebraic map between two
adjacent Uhlenbeck compactifications which restricts to the identity on some
Zariski open subset.


title: Riemann reciprocity in higher dimensions

abstract:
  The reciprocity law for abelian differentials of first and second kind is
generalized to higher-dimensional varieties. It is shown that $H^1(V)$ of a
polarized variety $V$ is encoded in the Laurent data along a curve germ in $V$,
with the polarization form on $H^1(V)$ corresponding to the {\em
one-dimensional} residue pairing. This associates an {\em extended abelian
variety} to $V$; if $V$ is an abelian variety itself, our construction
``extends" it, even when $V$ is not a Jacobian.


title:
Non-symmetric orthogonal geometry of Grothendieck rings of coherent
  sheaves on projective spaces

abstract:
  In this paper we consider orthogonal geometry of the free
$\protect\ZZ$-module $K_0(\protect\PP_n)$ with respect to the non-symmetric
unimodular bilinear form $$\chi(E,F)=\sum (-1)^\nu\dim\ext^\nu(E,F). $$ We
calculate the isometry group of this form and describe invariants of its
natural action on $K_0(\protect\PP_n)$. Also we consider some general
constructions with non-symmetric unimodular forms. In particular, we discuss
orthogonal decomposition of such forms and the action of the braid group on a
set of semiorthonormal bases. We formulate a list of natural arithmetical
conjectures about semiorthogonal bases of the form $\chi$.


title: Counting twisted cubic curves on general complete intersections

abstract:
  This paper has been rendered obsolete by our newer eprint alg-geom/9411005
"Bott's formula and enumerative geometry", which is a considerably expanded
version of the same paper, in spite of the change of titles. Please download
alg-geom/9411005 instead, and update possible references.


title: Semistable sheaves on Del Pezzo surfaces and Kronecker modules

abstract:
  In this paper we deal with semistable sheaves which can be represented as the
cokernel of an injective (or as the kernel of a surjective) morphism
$E_1\otimes\CC^m\longrightarrow E_2\otimes\CC^n$ , where $E_1$ and $E_2$ are
exceptional bundles. To each such a sheaf we assign the linear map
$\CC^m\otimes\hom^*(E_1,E_2)\longrightarrow \CC^n$. We obtain sufficient
conditions on the topological invariants of the sheaves for the moduli space of
the sheaves to be isomorphic to the quotient of an open subset in $\PP{\cal
L}(\CC^m\otimes\hom^*(E_1,E_2),\CC^n)$ under the action of
SL$(\CC^m)\times$SL$(\CC^n)$.


title:
Chamber structure of polarizations and the moduli of stable sheaves on a
  ruled surface

abstract:
  In this paper we shall generalize the chamber structure of polarizations
defined by Qin, and as an application we shall compute the Picard groups of
moduli spaces of stable sheaves on a non-rational ruled surface.


title:
Strong McKay Correspondence, String-theoretic Hodge Numbers and Mirror
  Symmetry

abstract:
  In the revised version of the paper, we correct misprints and add some new
statements.


title: Topology of Conjugate Varieties

abstract:
  Serre and Abelson have produced examples of non-homeomorphic conjugate
varieties. We show that if the field of definition of a polarized projective
variety coincides with its field of moduli then all of its conjugates have the
same topological type. This extends the class of varieties known to posses
conjugacy invariant to canonically embedded varieties. We also show that normal
complete interswections in homogeneous varieties have this property.


title: Moduli spaces $M_{g,n}(W)$ for surfaces

abstract:
  We construct and prove the projectiveness of the moduli spaces which are
natural generalizations to the case of surfaces of the following:
  1) $M_{g,n}$, the moduli space of $n$-marked stable curves,
  2) $M_{g,n}(W)$, the moduli space of $n$-marked stable maps to a variety $W$.


title: Moduli of high rank Vector bundles over Surface

abstract:
  This is the revised version of our previous preprint. In this paper, we
establish a generic smoothness result for moduli space of semistable sheaves of
arbitrary rank over surfaces provided that the second Chern class of the
sheaves is sufficiently large. The theorem is proved using degeneration theory.


title:
Variation of moduli spaces and Donaldson invariants under change of
  polarization

abstract:
  The paper determines the change of moduli spaces of rank $2$ sheaves on
surfaces with $p_g=0$ under change of polarization and the corresponding change
of the Donaldson invariants. In this revised version we have made some minor
stylistic changes in the previous text. In addition we have added a final
chapter of about 20 pages (announced in the previous version), in which the six
lowest order terms (three of them non-zero) of the change are computed
explicitely using computations in the cohomology of Hilbert schemes of points.


title:
Automorphisms and moduli spaces of varieties with ample canonical class
  via deformations of abelian covers

abstract:
  By a recent result of Viehweg, projective manifolds with ample canonical
class have a coarse moduli space, which is a union of quasiprojective
varieties. In this paper, we prove that there are manifolds with ample
canonical class that lie on arbitrarily many irreducible components of the
moduli; moreover, for any finite abelian group $G$ there exist infinitely many
components $M$ of the moduli of varieties with ample canonical class such that
the generic automorphism group $G_M$ is equal to $G$. In order to construct the
examples, we use abelian covers, i.e. Galois cover whose Galois group is finite
and abelian. We prove two results about abelian covers: first, that if the
building data are sufficiently ample, then the natural deformations surject on
the Kuranishi family of $X$; second, that if the building data are sufficiently
ample and generic, then $Aut(X)=G$.


title:
Flips of moduli spaces and transition formulas for Donaldson polynomial
  invariants of rational surfaces

abstract:
  We study the change of moduli spaces of Gieseker-semistable torsion free
rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the
surfaces are rational with an effective anti-canonical divisor, the moduli
spaces are linked by a series of flips (blowups and blowdowns). Using these
results, we compute the transition formulas for Donaldson polynomial invariants
of rational surfaces. Part of the work is also obtained independently by
Matsuki-Wentworth and Ellingsrud-G{\" o}ttsche.


title:
Applications of iterated curve blowup to set-theoretic complete
  intersections in P3

abstract:
  Let S, T be surfaces in P3. Suppose that S intersect T is set-theoretically a
smooth curve C of degree d and genus g. Suppose that S and T have no common
singular points. Then if C is not a complete intersection, then deg(S), deg(T)
< 2d^4. Fixing (d,g), one can form a finite (shorter) list of all possible
pairs (deg(S),deg(T)). For instance, when (d,g) = (4,0), and assuming for
simplicity that deg(S) <= deg(T):
  (deg(S), deg(T)) \in {(3,4), (3,8), (4,4), (4,7), (6,26), (9,48), (10,28)
  (12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33)}.
Assume characteristic 0. [1] Suppose that S and T have non-overlapping rational
singularities. Then d <= g+3. [2] Suppose that S is normal, and that d>deg(S).
Then C is linearly normal (and so d <= g+3). [3] Suppose that S is a quartic
surface having only rational singularities. Then C is linearly normal. Hard
copy is available from the author. E-mail to jaffe@cpthree.unl.edu.


title: Coherent functors, with application to torsion in the Picard group

abstract:
  Let A be a commutative noetherian ring. Call a functor <<commutative
A-algebras>> --> <<sets>> coherent if it can be built up (via iterated finite
limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g.
A-module. When such a functor F in fact takes its values in <<abelian groups>>,
we show that there are only finitely many prime numbers p such that _p F(A) is
infinite, and that none of these primes are invertible in A. This (and related
statements) yield information about torsion in Pic(A). For example, if A is of
finite type over Z, we prove that the torsion in Pic(A) is supported at a
finite set of primes, and if _p Pic(A) is infinite, then the prime p is not
invertible in A. These results use the (already known) fact that if such an A
is normal, then Pic(A) is finitely generated. We obtain a parallel result for a
reduced scheme X of finite type over Z. We show that the groups which can occur
as the Picard group of a scheme of finite type over a finite field all have the
form (finitely generated) + sum_{n=1}^infty F, where F is a finite p-group.
Hard copy is available from the author. E-mail to jaffe@cpthree.unl.edu.


title: Functorial structure of units in a tensor product

abstract:
  We study the units in a tensor product of rings. For example, let k be an
algebraically closed field. Let A and B be reduced rings containing k, having
connected spectra. Let u \in A tensor_k B be a unit. Then u = a tensor_k b for
some units a \in A and b \in B. Here is a deeper result, stated for simplicity
in the affine case only. Let k be a field, and let f: R --> S be a homomorphism
of f.g. k-algebras such that Spec(f) is dominant. Assume that every irreducible
component of Spec(R_red) or Spec(S_red) is geometrically integral and has a
rational point. Let B --> C be a faithfully flat homomorphism of reduced
k-algebras. For A a k-algebra, define Q(A) to be (S tensor_k A)^*/(R tensor_k
A)^*. Then Q satisfies the following sheaf property: the sequence
  0 --> Q(B) --> Q(C) --> Q(C tensor_B C)
  is exact. This and another result are used in the proof of the following
statement from "The kernel of the map on Picard groups induced by a faithfully
flat homomorphism" by R. Guralnick, D. Jaffe, W. Raskind, R. Wiegand:
  Let K/k be an algebraic field extension and let A be a f.g. k-algebra. Assume
resolution of singularities. Then there is a finite extension E/k contained in
K/k such that Pic(A tensor_k E) --> Pic(A tensor_k K) is injective.


title: Ample Line Bundles on Blown Up Surfaces

abstract:
  Given a smooth complex projective surface $S$ and an ample divisor $H$ on
$S$, consider the blow up of $S$ along $k$ points in general position. Let $H'$
be the pullback of $H$ and $E_1,..., E_k$ be the exceptional divisors. We show
that $L = nH' -E_1 - ... -E_k$ is ample if and only if $L^2$ is positive
provided the integer $n$ is at least 3.


title:
The principle describing possible combinations of singularities in
  deformations of a fixed singularity

abstract:
  This is the abstract prepared for Workshop on Topology and Geometry (Zhang
jiang, China, October 1994), and is a review of my recent works. What kinds of
combinations of singularities can appear in small deformation fibers of a fixed
singularity? We consider this problem for hypersurface singularities on complex
analytic spaces of dimension 2. For all singularities in the beginning par t of
Arnold's classification list, the answer to this problem is given by a unique
principle described by Dynkin graphs. This article contains several figures. I
will send the hard copy containin g figures by mail with envelope and stamps
under request.


title: On the Malcev completion of K\"ahler groups

abstract:
  Restrictions are placed on the presentation and holonomy algebra of a
K\"ahler group. Let $X$ be compact K\"ahler. It is known that the formality of
$X$, its Albanese map and the Hodge structure of $H^{\ast}(X)$ impose
restrictions on the holonomy algebra of $\pi_1 X$. We derive further
restrictions from Sullivan's 1-minimal model. An effective algorithm is given
to compute the holonomy algebra of a group from a presentation. Some very
strong restrictions on K\"ahler groups with few defining relations are
obtained.


title: On the linearization of the automorphism groups of algebraic domains

abstract:
  Let $D$ be a domain in $C^n$ and $G$ a topological group which acts
effectively on $D$ by holomorphic automorphisms. In this paper we are
interested in projective linearizations of the action of $G$, i.e. a linear
representation of $G$ in some $C^{N+1}$ and an equivariant imbedding of $D$
into $\P^N$ with respect to this representation. The domains we discuss here
are open connected sets defined by finitely many real polynomial inequalities
or connected finite unions of such sets. Assume that the group $G$ acts by
birational automorphisms.
  Our main result is the equivalence of the following conditions:
  1) there exists a projective linearization, i.e. a linear representation of
$G$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow
\P^N$ such that the restriction $i|_D$ is $G$-equivariant.
  2) $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms of
$D$ which extends the action of $G$ and has finitely many connected components;
  3) $G$ is a subgroup of a Nash group $\hat G$ of birational automorphisms of
$D$ which extends the action of $G$ to a Nash action $\hat G\times D\to D$;
  4) $G$ is a subgroup of a Nash group $\hat G$ such that the action $G\times
D\to D$ extends to a Nash action $\hat G\times D\to D$;
  5) the degree of the automorphism $\phi_g\colon D\to D$


title: On the Hitchin System

abstract:
  The Hitchin system is a completely integrable hamiltonian system (CIHS) on
the cotangent space to the moduli space of semi-stable vector bundles over a
curve. We consider the case of rank-two vector bundles with trivial
determinant. Such a bundle $E$ defines a divisor $D_E$ in the Jacobian of the
curve and for any smooth point of $D_E$ we define a cotangent vector (a Higgs
field). The Hitchin map on these Higgs fields is then determined in terms of
the Gauss map on the divisor $D_E$. We apply the results to the $g=2$ case and
show how Hitchin's system is related to classical line geometry in $\PP^3$.


title:
Mumford-Thaddeus Principle on the Moduli Space of Vector Bundles on an
  Algebraic Surface

abstract:
  We study the behavior of the Gieseker space of semistable torsion-free
sheaves of rank r and fixed c_1, c_2 on a non-singular projective surface as
the polarization varies. It is shown that the ample cone admits a locally
finite chamber structure, and that passing a wall adjacent to a pair of
chambers has the effect of modifying the moduli space by a (finite) sequence of
flips of the type studied by Thaddeus. The key steps are a modification of
Simpson's method and the introduction of a "rationally twisted" moduli space.
The result is more general but less explicit than the recent work of
Ellingsrud-Goettsche (alg-geom/9410005) and Friedman-Qin (alg-geom/9410007).


title: Toric Residues

abstract:
  In ${\bf C}^{n+1}$, one can show that the residue of $n+1$ homogeneous forms
of the same degree equals the integral of a certain $(n,n)$ form over ${\bf
P}^n$. Furthermore, the Jacobian of the forms has nonzero residue equal to a
certain intersection number. In this paper, we generalize these results to an
arbitrary projective toric variety. In particular, we define a toric version of
the Jacobian and show that it has the correct properties, and we give various
integral formulas for the toric residue. We also review some commutative
algebra results of Batyrev and Danilov and discuss the relation between the
trace map and the Dolbeault isomorphism.


title: A Matsusaka-type Theorem for Surfaces

abstract:
  The purpose of this note is to give an (esentially optimal) effective version
of Matsusaka's Big theorem for smooth projective surfaces.


title:
Recursive relations for the cohomology ring of moduli spaces of stable
  bundles

abstract:
  The authors learnt that similar results have been independently found by
D.Zagier, V.Baranovsky and V.Balaji/A.King/P.Newstead. The corresponding
references have been added (and some typos corrected).


title:
On the prime-to-$p$ part of the groups of connected components of
  N\'eron models

abstract:
  This article improves certain results of Dino Lorenzini concerning the groups
of connected components of special fibres of N\'eron models of abelian
varieties. Lorenzini has shown the existence of a four step filtration on the
prime-to-$p$ part ($p$ is the residue characteristic), and proved certain
bounds for the successive quotients. We improve these bounds and show that our
bounds are sharp. As an application, we give a complete classification of the
groups that can arise as the prime-to-$p$ part of the group of connected
components of the Neron model of an abelian variety with given dimensions for
the abelian, toric and unipotent part. Hard copies of this preprint are
available.


title:
The Theta Divisor of $SU_C(2,2d)^s$ is very Ample if $C$ is not
  Hyperelliptic

abstract:
  Let $X$ be the moduli space of semistable rank 2 vector bundles over a smooth
curve C of genus $g \ge 2$ and $\theta : X \to PH^0(L)^*$ be the map associated
to the generalized theta divisor L on X. We prove that for C not hyperelliptic,
the map $\theta$ is injective and the differential of $\theta$ is injective at
smooth points of X.


title:
$L^2$ vanishing theorems for positive line bundles and adjunction
  theory, Lecture Notes of a CIME course on "Transcendental Methods of
  Algebraic Geometry" (Cetraro, Italy, July 1994)

abstract:
  The notes start with an elementary introduction to a few important analytic
techniques of algebraic geometry: closed positive currents, $L^2$ estimates for
the $\dbar$-operator on positive vector bundles, Nadel's vanishing theorem for
multiplier ideal sheaves (a generalization of the well-known Kawamata-Viehweg
vanishing theorem). Applications to adjoint line bundles are then discussed.
T.~Fujita conjectured in 1987 that $K_X+(n+2)L$ is very ample for every ample
line bundle $L$ on a non singular projective variety $X$ with $\dim X=n$. The
answer is known only for $n\le 2$ (I.~Reider, 1988). In the last years, various
bounds have been obtained for integers $m$ such that $2K_X+mL$ is very ample
(by J.~Koll\'ar, L.~Ein-R.~Lazarsfeld, Y.T.~Siu and the author, among others).
Two approaches are discussed: an analytic approach via Monge-Amp\`ere equations
and current theory, and a more algebraic one (due to Siu) via multiplier ideal
sheaves and Riemann-Roch. Finally, an effective version of the big Matsusaka
theorem is derived, in the form of an explicit bound for an integer $m$ such
that $mL$ is very ample, depending only on $L^n$ and $L^{n-1}\cdot K_X$; these
bounds improve Siu's results (1993), and essentially contain the optimal bounds
obtained by Fernandez del Busto for the surface case.


title:
Remarks on rational points of varieties whose cotangent bundles are
  generated by global sections

abstract:
  In this short note, we will gives several remarks on rational points of
varieties whose cotangent bundles are generated by global sections. For
example, we will show that if the sheaf of differentials of a projective
variety X over a number field k is ample and generated by global sections, then
the set of k-rational points of X is finite.


title: Quantum Schubert Calculus

abstract:
  This paper works out the versions of the classical Giambelli and Pieri
formulas in the context of quantum cohomology of a complex Grassmannian.


title: On the automorphism groups of algebraic bounded domains

abstract:
  Let $D$ be a bounded domain in $C^n$. By the theorem of H.~Cartan, the group
$Aut(D)$ of all biholomorphic automorphisms of $D$ has a unique structure of a
real Lie group such that the action $Aut(D)\times D\to D$ is real analytic.
This structure is defined by the embedding $C_v\colon Aut(D)\hookrightarrow
D\times Gl_n(C)$, $f\mapsto (f(v), f_{*v})$, where $v\in D$ is arbitrary. Here
we restrict our attention to the class of domains $D$ defined by finitely many
polynomial inequalities. The appropriate category for studying automorphism of
such domains is the Nash category. Therefore we consider the subgroup
$Aut_a(D)\subset Aut(D)$ of all algebraic biholomorphic automorphisms which in
many cases coincides with $Aut(D)$. Assume that $n>1$ and $D$ has a boundary
point where the Levi form is non-degenerate. Our main result is theat the group
$Aut_a(D)$ carries a unique structure of an affine Nash group such that the
action $Aut_a(D)\times D\to D$ is Nash. This structure is defined by the
embedding $C_v\colon Aut_a(D)\hookrightarrow D\times Gl_n(C)$ and is
independent of the choice of $v\in D$.


title: Exceptional vector bundle on Enriques surfaces

abstract:
  The main purpose in this paper is to study exceptional vector bundles on
Enriques surfaces.


title:
Submodules of the deficiency modules and an extension of Dubreil's
  Theorem

abstract:
  In its most basic form, Dubreil's Theorem states that for an ideal $I$
defining a codimension $2$, arithmetically Cohen--Macaulay subscheme of
projective $n$-space, the number of generators of $I$ is bounded above by the
minimal degree of a minimal generator plus $1$. By introducing a new ideal $J$
which is the complete intersection of $n-1$ general linear forms, we are able
to extend Dubreil's Theorem to an ideal $I$ defining a locally Cohen--Macaulay
subscheme $V$ of any codimension. Our new bound involves the lengths of the
Koszul homologies of the cohomology modules of $V$, with respect to the ideal
$J$, and depends on a careful identification of the module $(I \cap J)/IJ$ in
terms of the maps in the free resolution of $J$. As a corollary to this
identification, we also give a new proof of a theorem of Serre which gives a
necessary and sufficient condition to have the equality $I \cap J = IJ$ in the
case where $I$ and $J$ define disjoint schemes in projective space.


title: Quantum Cohomology of Rational Surfaces

abstract:
  In this article formulas for the quantum product of a rational surface are
given, and used to give an algebro-geometric proof of the associativity of the
quantum product for strict Del Pezzo surfaces, those for which $-K$ is very
ample. An argument for the associativity in general is proposed, which also
avoids resorting to the symplectic category. The enumerative predictions of
Kontsevich and Manin concerning the degree of the rational curve locus in a
linear system are recovered. The associativity of the quantum product for the
cubic surface is shown to be essentially equivalent to the classical
enumerative facts concerning lines: there are $27$ of them, each meeting $10$
others.


title: Contractions on a manifold polarized by an ample vector bundle

abstract:
  A complex manifold $X$ of dimension $n$ together with an ample vector bundle
$E$ on it will be called a {\sf generalized polarized variety}. The adjoint
bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the
positivity (the nefness or ampleness) of the adjoint bundle in the case $r :=
rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of
paper by Ye-Zhang, Fujita, Andreatta-Ballico-Wisniewski.
  If $K_X+detE$ is nef, then by the Kawamata-Shokurov base point free theorem,
it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto
a normal projective variety $W$ with connected fiber and such that $K_X +
det(E) = \pi^*H$, for some ample line bundle $H$ on $W$. We describe those
contractions for which $dimF \leq (r-1)$. We extend this result to the case in
which $X$ has log terminal singualarities. In particular this gives the Mukai's
conjecture1 for singular varieties. We consider also the case in which $dimF =
r$ for every fibers and $\pi$ is birational. Hard copies of the paper are
available.


title: Self-linking number of a real algebraic link

abstract:
  For a nonsingular real algebraic curve in the 3-dimensional projective space
and sphere a new numeric characteristic is introduced. It takes integer values,
is invariant under rigid isotopy, multiplied by -1 under mirror reflection. In
a sense it is a Vassiliev invariant of degree 1 and a counter-part of a link
diagram writhe.


title: On the Picard group: torsion and the kernel

abstract:
  For a homomorphism f: A --> B of commutative rings, let D(A,B) denote
Ker[Pic(A) --> Pic(B)]. Let k be a field and assume that A is a f.g. k-algebra.
  We prove a number of finiteness results for D(A,B). Here are four of them.
  1: Suppose B is a f.g. and faithfully flat A-algebra which is geometrically
integral over k. If k is perfect, we find that D(A,B) is f.g. (In positive
characteristic, we need resolution of singularities to prove this.) For an
arbitrary field k of positive characteristic p, we find that modulo p-power
torsion, D(A,B) is f.g.
  2: Suppose B = A tensor k^sep. We find that D(A,B) is finite.
  3: Suppose B = A tensor L, where L is a finite, purely inseparable extension.
We give examples to show that D(A,B) may be infinite.
  4: Assuming resolution of singularities, we show that if K/k is any algebraic
extension, there is a finite extension E/k contained in K/k such that D(A
tensor E,A tensor K) is trivial.
  The remaining results are absolute finiteness results for Pic(A).
  5: For every n prime to char(k), Pic(A) has only finitely many elements of
order n.
  6: Structure theorems are given for Pic(A), in the case where k is absolutely
f.g.
  All of these results are proved in a more general form, valid for schemes.
Hard copy is available from the authors.


title: The geometry of A-graded algebras

abstract:
  We study algebras k[x_1,...,x_n]/I which admit a grading by a subsemigroup of
N^d such that every graded component is a one-dimensional k-vector space.
V.I.~Arnold and coworkers proved that for d = 1 and n <= 3 there are only
finitely many isomorphism types of such A-graded algebras, and in these cases I
is an initial ideal (in the sense of Groebner bases) of a toric ideal.
  In this paper it is shown that Arnold's finiteness theorem does not extend to
n = 4. Geometric conditions are given for I to be an initial ideal of a toric
ideal. The varieties defined by A-graded algebras are characterized in terms of
polyhedral subdivisions, and the distinct A-graded algebras are parametrized by
a certain binomial scheme.


title:
Towards a separation theorem of points by adjoint linear systems on
  polarized threefolds

abstract:
  Main Result:
  Let $(M,L)$ be a smooth complex polarized threefold. Then the linear system
$| K+tL|$ separates any two different points on $M$ for any $t\ge 6$, where $K$
is the canonical bundle of $M$. The argument in the proof is a variant of
Ein-Lazarsfeld method. Although it is less powerful, it is cheaper, i.e., needs
fewer pages. Unfortunately, at present, I cannot show the very ampleness
because of technical difficulties. Some related topics are also discussed. A
hard copy is available on request to the author.


title: The minimal number of singular fibers of a semistable curves over P^1

abstract:
  In this paper, we shall prove Beauville's conjecture: if $f:S \to P^1$ is a
non-trivial semistable fibration of genus g>1, then $f$ admits at least 5
singular fibers. We have also constructed an example of genus 2 with 5 singular
fibers. This paper will appear in the Journal of Algebraic Geometry.


title: On the invariants of base changes of pencils of curves, II

abstract:
  In this part of the series, we shall investigate Deligne-Mumford semistable
reductions from the point of view of numerical invariants. As an application,
we obtain two numerical criterions for a base change to be stabilizing, and for
a fibration to be isotrivial. We also obtain a canonical class inequality for
any fibration. Some other applications are presented. Most of the results of
this paper have arithmetical analogues. This paper will appear in Math. Z.


title: Operads and moduli spaces of genus $0$ Riemann surfaces

abstract:
  We study a pair of dual operads which arise in the study of moduli spaces of
pointed genus 0 curves (this duality is similar to that between commutative and
Lie algebras). These operads are both quadratic, and even Koszul, and arise in
the theory of quantum cohomology.


title: Bott's formula and enumerative geometry

abstract:
  We outline a strategy for computing intersection numbers on smooth varieties
with torus actions using a residue formula of Bott. As an example,
Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some
general complete intersection in projective space are computed. The results are
consistent with predictions made from mirror symmetry computations. We also
compute degrees of some loci in the linear system of plane curves of degrees
less than 10, like those corresponding to sums of powers of linear forms, and
curves carrying inscribed polygons.


title: Logarithmic Embeddings and Logarithmic Semistable Reductions

abstract:
  In this paper, we give a criterion for the existence of logarithmic
embeddings -- which was first introduced by Steenbrink -- for general normal
crossing varieties. Using this criterion, we also give a new proof of the
theorem of Kawamata--Namikawa which states a criterion for the existence of the
log structures of semistable type.


title: Conformal Differential Forms of an Analytic Germ

abstract:
  A differential form vanishing on the tangent space at smooth points of a
reduced embedded analytic germ is called conormal. For proving that a conormal
one--form of a hypersurface vanishes at its singularities we state a
Bertini--type theorem.


title: On the Corank of Gaussian Maps for General Embedded K3 Surfaces

abstract:
  Let $S_{g}$ be a general prime K3 surface in $P^g$ of genus $g \geq 3$ or a
general double cover of $P^2$ ramified along a sextic curve for $g = 2$ and $S
= S_{i,g}$ its {\it i}-th Veronese embedding. In this article we compute the
corank of the Gaussian map $\Phi:\bigwedge^2 H^0(S,O_S(1)) \to
H^0(S,\Omega_S^1(2))$ for $i \geq 2, g \geq 2$ and $i=1, g \geq 17$. The main
idea is to reduce the surjectivity of $\Phi$ to an application of the
Kawamata-Viehweg vanishing theorem on the blow-up of $S \times S$ along,the
diagonal. This is seen to apply once the hyperplane divisor of the K3 surface
$S$ can be decomposed as a sum of three suitable birationally ample divisors.
We show that such a decomposition exists when $i \geq 3$ or on some K3
surfaces, constructed using the surjectivity of the period mapping, when $i =
1, g \geq 17$ or $i=2, g \geq 7$.


title:
Rational Points of Bounded Height on Compactifications of Anisotropic
  Tori

abstract:
  We investigate the analytic properties of the zeta-function associated with
heights on equivariant compactifications of anisotropic tori over number
fields. This allows to verify conjectures about the distribution of rational
points of bounded height.


title: The McKay correspondence for finite subgroups of SL(3,\C)

abstract:
  This is the final draft, containing very minor proof-reading corrections. Let
G in SL(n,\C) be a finite subgroup and \fie: Y -> X = \C^n/G any resolution of
singularities of the quotient space. We prove that crepant exceptional prime
divisors of Y correspond one-to-one with ``junior'' conjugacy classes of G.
When n = 2 this is a version of the McKay correspondence (with irreducible
representations of G replaced by conjugacy classes). In the case n = 3, a
resolution with K_Y = 0 is known to exist by work of Roan and others; we prove
the existence of a basis of H^*(Y, \Q) by algebraic cycles in one-to-one
correspondence with conjugacy classes of G. Our treatment leaves lots of open
problems.


title:
Classification of Varieties with Canonical Curve Section via Gaussian
  maps on Canonical Curves

abstract:
  Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The
purpose of this article is to further develop a method to classify varieties
having $C$ as their curve section, using Gaussian map computations. In a
previous article a careful analysis of the degeneration to the cone over the
hyperplane section was made for _prime_ Fano threefolds, that is Fano
threefolds whose Picard group is generated by the hyperplane bundle. In this
article we extend this method and classify Fano threefolds of higher index
(which still have Picard number one). We are also able to classify Mukai
varieties, i.e. varieties of dimension four or more with canonical curve
sections.


title: On sextic surfaces having only nodes (preliminary report)

abstract:
  Let S be a surface in CP^3, having only nodes as singularities. Let pi: S~
--> S be a minimal resolution of singularities. A set N of nodes on S is EVEN
if there exists a divisor Q on S~ such that 2Q ~ pi^{-1}(N).
  Suppose that S has degree 6. It is known (Basset) that S cannot have 67 or
more nodes. It is also known (Barth) that S can have 65 nodes. It is not known
if S can have 66 nodes. Likewise, it is not known exactly what sizes can occur
for an even set of nodes on S.
  We show that an nonempty even set of nodes on S must have size 24, 32, 40,
56, or 64. We do not know if the sizes 56 and 64 can occur. We show that if S
has 66 nodes, then it must have an even set of 64 nodes, and it cannot have an
even set of 56 nodes. THUS IF ONE COULD RULE OUT THE CASE OF A 64 NODE EVEN
SET, IT WOULD FOLLOW THAT S CANNOT HAVE 66 NODES.
  The existence or nonexistence of large even node sets is related to the
following vanishing problem. Let S be a normal surface of degree s in CP^3. Let
D be a Weil divisor on S such that D is Q-rationally equivalent to rH, for some
r \in \Q. Under what circumstances do we have H^1(O_S(D)) = 0? For instance,
this holds when r < 0. For s=4 and r=0, H^1 can be nonzero. For s=6 and r=0, if
a 56 or 64 node even set exists, then H^1 can be nonzero. The vanishing of H^1
is also related to linear normality, quadric normality, etc. of set-theoretic
complete intersections in P^3. Hard copy is available from the author. E-mail
to jaffe@cpthree.unl.edu.


title:
A New Approach to Characterise all the Transitive Orientations for an
  Undirected Graph

abstract:
  A new approach to find all the transitive orientations for a comparability
graph (finite or infinite) is presented. This approach is based on the link
between the notion of ``strong'' partitive set and the forcing theory (notions
of simplices and multiplices). A mathematical algorithm is given for the case
of a comparability graph which has only non limit sub-graphs.


title: A Borel-Weil Theorem for Schur Modules

abstract:
  We present a generalization of the classical Schur modules of $GL(N)$
exhibiting the same interplay among algebra, geometry, and combinatorics. A
generalized Young diagram $D$ is an arbitrary finite subset of $\NN \times
\NN$. For each $D$, we define the Schur module $S_D$ of $GL(N)$. We introduce a
projective variety $\FF_D$ and a line bundle $\LL_D$, and describe the Schur
module in terms of sections of $\LL_D$. For diagrams with the ``northeast''
property,
  $$(i_1,j_1),\ (i_2, j_2) \in D \to (\min(i_1,i_2),\max(j_1,j_2)) \in D ,$$
  which includes the skew diagrams, we resolve the singularities of $\FD$ and
show analogs of Bott's and Kempf's vanishing theorems. Finally, we apply the
Atiyah-Bott Fixed Point Theorem to establish a Weyl-type character formula of
the form:
  $$ {\Char}_{S_D}(x) = \sum_t {x^{\wt(t)} \over \prod_{i,j} (1-x_i
x_j^{-1})^{d_{ij}(t)}} \ ,$$
  where $t$ runs over certain standard tableaux of $D$.
  Our results are valid over fields of arbitrary characteristic.


title: Uniformity of rational points over all quadratic fields

abstract:
  We refine a result of L. Caporaso, J. Harris and B. Mazur, and prove:
Supposons que la conjecture de Lang soit vraie. Soit $K$ un corps des nombres
et $g>1$ un entier. Il existe un nombre $N(K,g)$ tel que si $L$ est une
extension de degr\'e $\leq 3$ de $K$ et $C$ est une courbe lisse projective
conn\`exe, de genre $g$ d\'efinie sur $L$ on a $$\# C(L) < N(K,g). $$


title: Bounding the order of automorphisms of certain curves

abstract:
  We study upper bounds on the order of automorphisms of non-singular curves
$X$ satisfying at least one of the following hypothesis: 1) $X$ is an
$m$-sheeted covering of exactly one non-singular curve of genus $\gamma$, where
$m$ is prime; 2) the center of the group of automorphisms of $X$ is
non-trivial.


title: Prym Varieties and the Moduli of Spin Bundles

abstract:
  We show that the Verlinde formula for moduli spaces of spin bundles on an
algebraic curve gives dimensions of direct sums of spaces of theta functions
over the finite set of Prym varieties of unramified double covers of the curve.
We then construct a natural duality (though we do not show it is nondegenerate)
between the latter spaces and the spaces of sections of the theta bundles on
the moduli spaces of spin bundles corresponding to the orthogonal
representation. Finally, it is remarked that this should be viewed as a special
case of a conjectured `strange duality' for spin bundles.


title: Homological Algebra of Mirror Symmetry

abstract:
  This is my talk at ICM, Zurich 1994. It contains a short introduction, two
basic examples and a refined version of the Mirror Conjecture formulated in
terms of homological algebra.


title: Mirror symmetry and moduli spaces of superconformal field theories

abstract:
  We review the applications of mirror symmetry to the study of the moduli
spaces of two-dimensional conformal field theories with $N{=}(2,2)$
supersymmetry, particularly those constructed from Calabi--Yau manifolds.
(Lecture delivered at the Zurich ICM, section on mathematical physics.)


title: Higher $l$-adic Abel-Jacobi mappings and filtrations on Chow groups

abstract:
  We define a filtration on the Chow groups of a smooth projective variety X
over a field k by using the cycle map into continuous l-adic etale cohomology.
The main theorem says that if k is a function field in one variable over a
finite field, this filtration for zero cycles is of length at most one modulo
the kernel of the cycle map.


title: Cellular decompositions of compactified moduli spaces of pointed curves

abstract:
  To a closed connected oriented surface $S$ of genus $g$ and a nonempty finite
subset $P$ of $S$ is associated a simplicial complex (the arc complex) that
plays a basic r\^ ole in understanding the mapping class group of the pair
$(S,P)$. It is known that this arc complex contains in a natural way the
product of the Teichm\"uller space of $(S,P)$ with an open simplex. In this
paper we give an interpretation for the whole arc complex and prove that it is
a quotient of a Deligne--Mumford extension of this Teichm\"uller space and a
closed simplex. We also describe a modification of the arc complex in the
spirit of Deligne--Mumford.


title: A lecture on Kac--Moody Lie algebras of the arithmetic type

abstract:
  We name an indecomposable symmetrizable generalized Cartan matrix $A$ and the
corresponding Kac--Moody Lie algebra ${\goth g} ^\prime (A)$ {\it of the
arithmetic type} if for any $\beta \in Q$ with $(\beta | \beta)<0$ there exist
$n(\beta )\in {\Bbb N}$ and an imaginary root $\alpha \in \Delta^{im}$ such
that $n(\beta )\beta \equiv \alpha \mod Ker\ (.|.)$ on $Q$. Here $Q$ is the
root lattice. This generalizes "symmetrizable hyperbolic" type of Kac and
Moody. We show that generalized Cartan matrices of the arithmetic type are
divided in $4$ types: (a) finite, (b) affine, (c) rank two, and (d) arithmetic
hyperbolic type. The last type is very closely related with arithmetic groups
generated by reflections in hyperbolic spaces with the field of definition
$\Bbb Q$. We apply results of the author and \'E.B. Vinberg on arithmetic
groups generated by reflections in hyperbolic spaces to describe generalized
Cartan matrices of the arithmetic hyperbolic type and to show that there exists
a finite set of series of the generalized Cartan matrices of the arithmetic
hyperbolic type. For the symmetric case all these series are known.


title:
Fulton-Hansen and Barth-Lefschetz Theorems for Subvarieties of Abelian
  Varieties

abstract:
  This is the published (Crelles Journal 467 (1995), 187--197) version of an
e-print entitled "ON SUBVARIETIES OF ABELIAN VARIETIES".


title: Sous-vari\'et\'es de codimension $2$ d'une vari\'et\'e ab\'elienne

abstract: This is the published version. Previous version had minor errors. In French.


title: On the Deligne--Beilinson cohomology sheaves

abstract:
  We are showing that the Deligne--Beilinson cohomology sheaves ${\cal
H}^{q+1}({\bf Z}(q)_{\cal D})$ are torsion free by assuming Kato's conjectures
hold true for function fields. This result is `effective' for $q=2$; in this
case, by dealing with `arithmetic properties' of the presheaves of mixed Hodge
structures defined by singular cohomology, we are able to give a cohomological
characterization of the Albanese kernel for surfaces with $p_g=0$.


title: Characteristic classes and quadric bundles

abstract:
  We construct Euler and Stiefel-Whitney classes of vector bundles with
quadratic form by analyzing the intersection theory of the associated quadric
bundles. We also compute the Chow rings of quadric and isotropic flag bundles.
Along the way, we prove a conjecture of Fulton on top Chern classes of maximal
isotropic sub-bundles of an even rank quadratic vector bundle.


title: Characteristic classes in the Chow ring

abstract:
  We study the ring of characteristic classes with values in the Chow ring for
principal $G$-bundles over schemes. If we consider bundles which are locally
trivial in the Zariski topology, then we show, for $G$ reductive, that this
ring is isomorphic to the Weyl group invariants in the algebra generated by
characters of the maximal torus. For general principal bundles the same
isomorphism holds after tensoring the coefficients with ${\Bbb Q}$. As a
corollary, we show that any (non-torsion) topological characteristic class is
algebraic when applied to Zariski locally trivial bundles over complex
algebraic varieties.


title:
$\beta$-nbc bases for cohomology of local systems on hyperplane
  complements

abstract:
  We study cohomology with coefficients in a rank one local system on the
complement of an arrangement of hyperplanes $\A$. The cohomology plays an
important role for the theory of generalized hypergeometric functions. We
combine several known results to construct explicit bases of logarithmic forms
for the only non-vanishing cohomology group, under some nonresonance conditions
on the local system, for any arrangement $\A$. The bases are determined by a
linear ordering of the hyperplanes, and are indexed by certain
``no-broken-circuits" bases of $\A$. The basic forms depend on the local
system, but any two bases constructed in this way are related by a matrix of
integer constants which depend only on the linear orders and not on the local
system. In certain special cases we show the existence of bases of monomial
logarithmic forms.


title:
The multiple-point schemes of a finite curvilinear map of codimension
  one

abstract:
  Let X and Y be smooth varieties of dimensions n-1 and n over an arbitrary
algebraically closed field, f:X-> Y a finite map that is birational onto its
image. Suppose that f is curvilinear; that is, at every point of X, the
Jacobian has rank at least n-2. For r at least 1, consider the subscheme N_r of
Y defined by the (r-1)st Fitting ideal of the O_Y-module f_*O_X, and set
M_r:=f^{-1}N_r. In this setting --- in fact, in a more general setting --- we
prove the following statements, which show that M_r and N_r behave like
reasonable schemes of source and target r-fold points of f.
  Each component of M_r and N_r is empty or has dimension at least n-r. If each
component of M_r, or equivalently of N_r, has dimension n-r, then M_r and N_r
are Cohen--Macaulay, and their fundamental cycles satisfy the relation,
f_*[M_r]=r[N_r]. Now, suppose that each component of M_s, or of N_s, has
dimension n-s for s=1,...,r+1. Then the blowup Bl(N_r,N_{r+1}) is equal to the
Hilbert scheme Hilb^r_f, and the blowup Bl(M_r,M_{r+1}) is equal to the
universal subscheme Univ^r_f of Hilb^r_f x_Y X; moreover, Hilb^r_f and Univ^r_f
are Gorenstein. In addition, the structure map h:Hilb^r_f->Y is finite and
birational onto its image; and its conductor is equal to the ideal J_r of
N_{r+1} in N_r, and is locally self-linked. Reciprocally, h_*O_{Hilb^r_f} is
equal to Hom(J_r,O_{N_{r}}). Moreover,
  h_*[h^{-1}N_{r+1}]=(r+1)[N_{r+1}].
  Furthermore, similar assertions hold for the structure map h_1:Univ^r_f->X if
r>1.


title:
Quadratic forms, generalized Hamming weights of codes and curves with
  many points

abstract:
  We use the relations between quadrics, trace codes and algebraic curves to
construct algebraic curves over finite fields with many points and to compute
generalized Hamming weights of codes.


title:
On Degenerate Secant and Tangential Varieties and Local Differential
  Geometry

abstract:
  We study the local differential geometry of varieties $X^n\subset \Bbb C\Bbb
P^{n+a}$ with degenerate secant and tangential varieties. We show that the
second fundamental form of a smooth variety with degenerate tangential variety
is subject to certain rank restrictions. The rank restrictions imply a slightly
refined version of Zak's theorem on linear normality and a short proof of the
Zak-Fantecchi theorem on the superadditivity of multisecant defects. We show
there is a vector bundle defined over general points of $TX$ whose fibers carry
the structure of a Clifford algebra. This structure implies additional
restrictions of the size of the secant defect. The Clifford algebra structure,
combined with further local computations, yields a new proof of Zak's theorem
on Severi varieties that is substantially shorter than the original. We also
prove local and global results on the dimension of the Gauss image of
degenerate tangential varieties, refining the results in [GH].


title: Even Linkage Classes

abstract:
  In this paper the author generalizes the $\E$ and $\N$-type resolutions used
by Martin-Deschamps and Perrin to subschemes of pure codimension in projective
space, and shows that these resolutions are interchanged by the mapping cone
procedure under a simple linkage. Via these resolutions, Rao's correspondence
is extended to give a bijection between even linkage classes of subschemes of
pure codimension two and stable equivalence classes of reflexive sheaves $\E$
satisfying $H^1_*(\E)=0$ and $\ext^1(\E^\vee, \O)=0$. Further, these
resolutions are used to extend the work of Martin-Deschamps and Perrin for
Cohen-Macaulay curves in $\Pthree$ to subschemes of pure codimension two in
$\Pn$. In particular, even linkage classes of such subschemes have the
Lazarsfeld-Rao property and any minimal subscheme for an even linkage class is
directly linked to a minimal subscheme for the dual class.


title:
Hilbert schemes of points on some K3 surfaces and Gieseker stable
  bundles

abstract:
  By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that
for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\Hilb^n(X)$
can be identified, for all $n\geq 1$, with moduli spaces of Gieseker stable
vector bundles on $X$ of rank $1+2n$. We also introduce a new Fourier-Mukai
type transform for such surfaces.


title:
On $CR$ mappings between algebraic Cauchy--Riemann manifolds and
  separate algebraicity for holomorphic functions

abstract:
  We prove the algebraicity of smooth $CR$-mappings between algebraic
Cauchy--Riemann manifolds. A generalization of separate algebraicity principle
is established.


title: On the Hilbert Function of Fat Points on a Rational Normal Cubic

abstract:
  In this paper we find an algorithm which computes the Hilbert function of
schemes $Z$ of "fat points" in $\PP3$ whose support lies on a rational normal
cubic curve $C$. The algorithm shows that the maximality of the Hilbert
function in degree $t$ is related to the existence of fixed curves (either $C$
itself or one of its secant lines) for the linear system of surfaces of degree
$t$ containing $Z$.


title: On Calabi-Yau Complete Intersections in Toric Varieties

abstract:
  We investigate Hodge-theoretic properties of Calabi-Yau complete
intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano
varieties having at most Gorenstein singularities. Our main purpose is to show
that the combinatorial duality proposed by second author agrees with the
duality for Hodge numbers predicted by mirror symmetry. It is expected that the
complete verification of mirror symmetry predictions for singular Calabi-Yau
varieties $V$ of arbitrary dimension demands considerations of so called {\em
string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict ourselves to
the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm
st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers
$h^{0,q}(\widehat{V})$ and $h^{1,q}(\widehat{V})$ of a $MPCP$-desingularization
$\widehat{V}$ of $V$.


title: Moduli of Polarised Abelian Surfaces

abstract:
  The toroidal compactification of the moduli space of complex abelian surfaces
with a polarisation of type (1,p), p a prime, is of general type if p is at
least 173. Happy Christmas.


title: Mixed Buchsbaum--Rim Multiplicities

abstract:
  We prove the results about mixed Buchsbaum--Rim multiplicities announced in
(9.10)(ii) on p.224 of our recent paper [J.Alg.(1994)], including a general
mixed-multiplicity formula. In addition, we identify these multiplicities as
the coefficients of the ``leading form'' of the appropriate Buchsbaum-Rim
polynomial in three variables, and we prove a positivity theorem. In fact, we
define the multiplicities as the degrees of certain zero-dimensional ``mixed
twisted'' Segre classes, and we develop an encompassing general theory of these
new rational equivalence classes in all dimensions. In parallel, we develop a
theory of pure ``twisted'' Segre classes, and we recover the main results in
[J.Alg.(1994)] about the pure Buchsbaum--Rim multiplicities, the polar
multiplicities, and so forth. Moreover, we identify the additivity theorem
[J.Alg.(1994), (6.7b)(i), p.205] as giving a sort of residual-intersection
formula, and we show its (somewhat unexpected) connection to the
mixed-multiplicity formula. Also, we work in a more general setup than before,
and we develop a new approach, based on the completed normal cone.


title:
An elementary approach to the abelianization of the Hitchin system for
  arbitrary reductive groups

abstract:
  We consider the moduli space of stable principal G-bundles over a compact
Riemann surface C of genus >1, with G a reductive algebraic group. We
explicitly construct a map F from the generic fibre of the Hitchin map to a
generalized Prym variety associated to a suitable Galois covering of C. The map
F has finite fibres. In case G=PGl(2) one can check that the generic fibre of F
is a principal homogeneous space with respect to a product of 2d-2 copies of
Z/2Z where d is the degree of the canonical bundle over C. However in case the
Dynkin diagram of G does not contain components of type $B_{n}$ n>0, or when
the commutator subgroup (G,G) is simply connected the map F is injective.


title: Remarks on S. Lang's conjecture over function fields

abstract:
  In this short note, we will show the following weak evidence of S. Lang
conjecture over function fields. Let f : X ---> Y be a projective and
surjective morphism of algebraic varieties over an algebraically closed field k
of characteristic zero, whose generic fiber is geometrically irreducible and of
general type. If f is not birationally trivial, then there are countably many
proper closed varieties { Z_i } of X such that every quasi-section of f is
contained in the union of Z_i.


title: Grothendieck duality via homotopy theory

abstract:
  Grothendieck proved that if $f:X\longrightarrow Y$ is a proper morphism of
nice schemes, then $Rf_*$ has a right adjoint, which is given as tensor product
with the relative canonical bundle. The original proof was by patching local
data. Deligne proved the existence of the adjoint by a global argument, and
Verdier showed that this global adjoint may be computed locally.
  In this article we show that the existence of the adjoint is an immediate
consequence of Brown's representability theorem. It follows almost as
immediately, by ``smashing'' arguments, that the adjoint is given by tensor
product with a dualising complex. Verdier's base change theorem is an immediate
consequence.


title: Cohomology of compact hyperkaehler manifolds

abstract:
  Let M be a compact simply connected hyperk\"ahler (or holomorphically
symplectic) manifold, \dim H^2(M)=n. Assume that M is not a product of
hyperkaehler manifolds. We prove that the Lie algebra so(n-3,3) acts by
automorphisms on the cohomology ring H^*(M). Under this action, the space
H^2(M) is isomorphic to the fundamental representation of so(n-3,3). Let A^r be
the subring of H^*(M) generated by H^2(M). We construct an action of the Lie
algebra so(n-2,4) on the space A, which preserves A^r. The space A^r is an
irreducible representation of so(n-2,4). This makes it possible to compute the
ring A^r explicitely.


abstract:
  Remarks on the Obstructedness of Cones Over Curves of Low Genus. Reason for
replacement: proof of the main lemma was significantly improved.


title: Moduli of curves with non-abelian level structure

abstract:
  Following Deligne and Mumford we construct a coarse moduli space of smooth
curves with non-abelian level structure, involving higher order commutators. We
prove that its Deligne-Mumford compactification is smooth over an open part of
Spec(${\msy Z}$).


title:
${\bf C}^*$-extensions of tori, higher Chow groups and applications to
  incidence equivalence relations for algebraic cycles.

abstract:
  Let X be a smooth projective variety of dimension n. If $p+q=n+1$ then Bloch
has defined a ${\bf G}_m$-biextension E over the product of the Chow groups
$CH^p_0(X)$ and $CH^q_0(X)$ of homologically trivial cycles. We prove that E is
the pullback of the Poincare biextension over the product of intermediate
Jacobians in characteristic zero. This is used to study various equivalence
relations for algebraic cycles. In particular we reprove Murres result that
Griffiths conjecture holds for codimension two cycles, i.e. every codim. two
cycle algebraically and incidence equivalent to zero has torsion Abel-Jacobi
invariant.


title:
An improved bound for the degree of smooth surfaces in P4 not of general
  type.

abstract:
  This is an addendum to the paper of Braun and Fl{\o}ystad ([BF]) on the bound
for the degree of a smooth surface in $\pfour$ not of general type. Using their
construction and the regularity of curves in $\pthree$, one may lower the bound
a little more. We will prove the following:
  Theorem. Let $S$ be a smooth surface of degree $d$ in $\pfour$ not of general
type. Then either $d \leq 70$ or $S$ lies on a hypersurface in $\pfour$ of
degree 5 and $d \leq 80$. Thus $d \leq 80$. We will also indicate how to lower
the bound to 76.


title: Boundary behaviour of Hurwitz schemes

abstract:
  Actions of finite groups on stable curves are studied. They appear naturally
at the boundary of a moduli space of smooth curves with group actions. Those
actions which can equivariantly smoothed are characterised. A description of
topological types of those actions in terms of the quotient curve and mappings
from a kind of fundamental group to the given finite group is given, analogous
to the well known case of actions on smooth curves.


title: Plane Curves with Hyperbolic and C-hyperbolic Complements

abstract:
  The general problem which initiated this work is:
  What are the quasiprojective varieties which can be uniformized by means of
bounded domains in $\cz^n$ ?
  Such a variety should be, in particular, C--hyperbolic, i.e. it should have a
Carath\'{e}odory hyperbolic covering. We study here the plane projective curves
whose complements are C--hyperbolic. For instance, we show that most of the
curves whose duals are nodal or, more generally, immersed curves, belong to
this class. We also give explicit examples of irreducible such curves of any
even degree d greater or equal 6.


title: Quantum cohomology of complete intersections

abstract:
  The quantum cohomology algebra of a projective manifold X is the cohomology
H(X,Q) endowed with a different algebra structure, which takes into account the
geometry of rational curves in X. We show that this algebra takes a remarkably
simple form for complete intersections when the dimension is large enough
w.r.t. the degree. As a reward we get a number of surprising enumerative
formulas relating lines, conics and twisted cubics on X.


title: On the Hilbert scheme of curves in higher-dimensional projective space

abstract:
  In this paper we prove that, for any $n\ge 3$, there exist infinitely many
$r\in \N$ and for each of them a smooth, connected curve $C_r$ in $\P^r$ such
that $C_r$ lies on exactly $n$ irreducible components of the Hilbert scheme
$\hilb(\P^r)$. This is proven by reducing the problem to an analogous statement
for the moduli of surfaces of general type.


title: On the tautological ring of $\M _g$

abstract:
  We prove among other things that any product of tautological classes of
$\M_g$ of degree $d$ (in the Chow ring of $\M _g$ with rational coefficients)
vanishes for $d>g-2$ and is proportional to the class of the hyperelliptic
locus in degree $g-2$.


title: Some homogeneous coordinate rings that are Koszul algebras

abstract:
  Using reduction to positive characteristic and the method of Frobenius
splitting of diagonals, due to Mehta and Ramanathan, it is shown that
homogeneous coordinate rings for either proper and smooth toric varieties or
Schubert varieties are Koszul algebras.


title: On the homogeneous ideal of a projective nonsingular toric variety

abstract:
  Reason for withdrawal:
  There is a serious mistake in the calculation of the divisor of the rational
section used in the proof of Prop. 2.2.1., and with the correct value the
argument does not work.


title: The motive of some moduli spaces of vector bundles over a curve

abstract:
  We study the motive of the moduli spaces of semistable rank two vector
bundles over an algebraic curve. When the degree is odd the moduli space is a
smooth projective variety, we obtain the absolute Hodge motive of this, and in
particular the Hodge-Poincare polynomial. When the degree is even the moduli
space is a singular projective variety, we compute pure Euler characteristics
and show that only two weights can occur in each cohomology group, we also see
that its cohomology is pure up to a certain degree. As a by-product we obtain
the isogeny type of some intermediate jacobians of the moduli spaces.


title: A sextic surface cannot have 66 nodes

abstract:
  Let S be a surface in complex projective 3-space, having only nodes as
singularities. Suppose that S has degree 6. We show that the maximum number of
nodes which S can have is 65.
  An abbreviated history of this is as follows. Basset showed that S can have
at most 66 nodes. Catanese and Ceresa and Stagnaro constructed sextic surfaces
having 64 nodes. Barth has recently exhibited a 65 node sextic surface. We
complete the story by showing that S cannot have 66 nodes.
  Let f: S~ --> S be a minimal resolution of singularities. A set N of nodes on
S is even if there exists a divisor Q on S~ such that 2Q ~ f^{-1}(N). We show
that a nonempty even set of nodes on S must have size 24, 32, 40, 56, or 64.
This result is key to showing the nonexistence of the 66 node sextic.
  We do not know if a sextic surface can have an even node set of size 56 or
64. The existence or nonexistence of large even node sets is related to the
following vanishing problem. Let S be a normal surface of degree s in CP^3. Let
D be a Weil divisor on S such that D is Q-rationally equivalent to rH, for some
r \in \Q. Under what circumstances do we have H^1(O_S(D)) = 0? For instance,
this holds when r < 0. For s=4 and r=0, H^1 can be nonzero. For s=6 and r=0, if
a 56 or 64 node even set exists, then H^1 can be nonzero. The vanishing of H^1
is also related to linear normality, quadric normality, etc. of set-theoretic
complete intersections in P^3.


title: A Generalization of the Kodaira Vanishing and Embedding Theorem

abstract:
  We give several generalizations of the Kodaira vanishing and embedding
theorems for K\"ahler manifolds to the case where the relevent line bundle has
a small region of negative curvature. To prove the vanishing theorems we adapt
techniques of Elworthy-Rosenberg for vanishing theorems in Riemannian geometry.
For the embedding theorem, we show that a K\"ahler manifold with a mostly
positive line bundle is Moishezon, since the usual blow up techniques do not
work in our situation.


title: A Calabi-Yau threefold with non-Abelian fundamental group

abstract: The title is self-explanatory.


title: Duality of Weights, Mirror Symmetry and Arnold's Strange Duality

abstract:
  A notion of duality of weight systems which corresponds to Batyrev's toric
mirror symmetry is given. Explicit duality on the (1,1)-cohomology of K3
surfaces which are minimal models of toric hypersurfaces is constructed using
monomial divisor mirror map of Aspinwall-Greene-Morrison. It is shown that
Arnold's strange duality for exceptional unimodal singularities reduces to this
duality.


title: Mirror symmetry for lattice polarized K3 surfaces

abstract:
  We establish a relationship between mirror symmetry for K3 surfaces and
Arnold's strange duality for K3 surfaces. We compute various examples of mirror
families. Among them the mirror moduli family for the moduli space of degree 2n
polarized K3 surfaces. It turns out to be related to the moduli space of
elliptic curves with level n.


title: Homology of moduli spaces of curves and commutative homotopy algebras

abstract:
  We propose an explicit relation between the cohomology of compactified and
noncompactified moduli spaces of algebraic curves with punctures. This
relationship generalizes one between commutative algebras and Lie algebras
proposed by Lazard, Kontsevich and Ginzburg and Kapranov. More explicitly, we
show that the cohomology of one moduli space is the graph complex decorated
with the cohomology of the other, which in its turn generalizes Getzler's
result for the genus zero moduli. We also show that a certain class of algebras
over the chain operad of compactified moduli spaces has a natural structure of
a commutative homotopy algebra. This may be regarded as a lifting of the dot
product of Lian-Zuckerman's BV algebra structure to the cochain level, thus
complementing the results of an earlier paper hep-th/9307114 of the authors
regarding the BV bracket.


title: Log Sarkisov Program

abstract:
  The purpose of this paper is two-fold. The first is to give a tutorial
introduction to the Sarkisov program, a 3-dimensional generalization of
Castelnuovo-N\"other Theorem ``untwisting" birational maps between Mori fiber
spaces, which was recently established by Corti after Sarkisov and Reid.
  The second is an attempt to give a logarithmic generalization of the program,
introducing the notion of the Sarkisov relation. We give a simple description
of the structure of the program and clarify the mechanism of termination
connected to the boundedness conjecture of (log) Q-Fano n-folds by Borisov.


title: On singularities of ${\cal M}_{I\!\! P^3}(c_1,c_2)$

abstract:
  Let ${\cal M}_{I\!\! P^3}(c_1,c_2)$ be the moduli space of stable rank-$2$
vector bundles on $I\!\! P^3$ with Chern classes $c_1$, $ c_2$. We prove the
following results.
  1) Let $0 \le \beta < \gamma $ be two integers, ($\gamma \ge 2)$, such that
$2\gamma-3\beta>0$; then ${\cal M}_{I\!\! P^3}(0,2\gamma^2-3\beta^2)$ is
singular (the case $\beta =0$ was previously proved by M. Maggesi).
  2) Let $0 \le \beta < \gamma $ be two odd integers ($\gamma \ge 5)$, such
that $2\gamma-3\beta+1>0$; then ${\cal M}_{I\!\!
P^3}(-1,2(\gamma/2)^2-3(\beta/2)^2+1/4)$ is singular.
  In particular ${\cal M}_{I\!\! P^3}(0,5)$, ${\cal M}_{I\!\! P^3}(-1,6)$ are
singular.


title:
Braid Group techniques in Complex Geometry V: The fundamental group of
  the complement of a Veronese generic projection

abstract:
  Computation of the fundamental group of the complement in the complex plane
of the branch curve S , of a generic projection of the Veronese surface to the
plane is presented. This paper is a continuation of our previous papers: Braid
Group Technique I - IV. In I and II we developed algorithms to compute braid
monodromy of a brunch curve, provided there exist a degeneration of the surface
to union of planes in a configuration where the associated branch curve is
partial to a line arrangement dual to generic. In III we constructed a
degeneration of the Veronese surface of order 3 to union of planes with the
desired property and in IV we used I -III to compute the braid monodromy of the
associated branch curve. Here we use the Van-Kampen method to compute the
fundamental group of the complement from a braid monodromy factorization and
some extra properties of the factorization , namely invariant properties, also
proven in IV. The group is presented using a certain quotient of the Braid
group defined by transversal half-twists.


title: Operads, Grothendieck topologies and Deformation theory

abstract:
  The idea of the work is to find an invariant way to pass from deformation
theory to cohomology, which does not use any explicit cocycles. The appropriate
cohomology theory is based on considering sheaves on a certain site. An
advantage of the approach is that it enables one to give a simultanious
treatement to all types of algebras. As an application, we prove the PBW
theorem in cases where it is not yet known.


title: Height inequality of algebraic points on curves over functional fields

abstract:
  The purpose of this paper is to give a linear and effective height inequality
for algebraic points on curves over functional fields. Our height inequality
can be viewed as the logarithmic canonical class inequality of a punctured
curve over a functional field (a fibered surface minus a section).


title: k-very ample line bundles on Del Pezzo Surfaces

abstract:
  A $k$-very ample line bundle L on a Del Pezzo Surface is numerically
characterized. We find that the set of the exceptional curves and effective
divisors with selfintersection zero, $\cal S$, plays a very important rule in
checking the nefness and $k$-very ampleness of a line bundle on a Del pezzo
Surface. We show that a line bundle L is nef if and only if it is spanned and
that L is $k$-very ample if and only if the intersection with all the elements
of $\cal S$ is greater or equal than k. In the first part of the paper we give
a complete numerical description of the elements of $\cal S$ .


title: Abelian automorphism groups of threefolds of general type

abstract:
  This thesis is devoted to the study of abelian automorphism groups of
surfaces and $3$-folds of general type over complex number field $\Bbb C$. We
obtain a linear bound in $K^3$ for abelian automorphism groups of $3$-folds of
general type whose canonical divisor $K$ is numerically effective, and we
improve on Xiao's results on abelian automorphism groups of minimal smooth
projective surfaces of general type. More precisely, the main results in this
thesis are the following. {\bf Theorem 3.0.} Let $X$ be a smooth 3-fold of
general type over the complex number field, $K$ the canonical divisor of $X$.
Let $G$ be an abelian group of automorphisms of $X$. Suppose $K$ is nef. Then
there exists a universal constant coefficient $c$ such that $\# G \le c K^3$.


title:
Self-intersection of dualizing sheaves of arithmetic surfaces with
  reducible fibers

abstract:
  Let K be an algebraic number field and O_K the ring of integers of K. Let f :
X --> Spec(O_K) be a stable arithmetic surface over O_K of genus g >= 2. In
this short note, we will prove that if f has a reducible geometric fiber, then
the self intersection of dualizing sheaf of X with Arakelov metric is greater
than or equal to log(2)/6(g-1).


title: Fundamental groups of complements of curves as solvable groups

abstract:
  We discuss the applications of fundamental groups (of complements of curves)
computations (and possibly the computations of the second homotopy group as a
model over it) to the classification of algebraic surface. We prove that the
fundamental group of the complement of the branch curve of a generic projection
of a Veronese surface to the complex plane is an "almost solvable" group in the
sense that it contains a solvable group of finite index and thus we can
consider the second fundamental group as model over the first.


title: Embeddings of homogeneous spaces in prime characteristics

abstract:
  Let $G$ be a reductive linear algebraic group. The simplest example of a
projective homogeneous $G$-variety in characteristic $p$, not isomorphic to a
flag variety, is the divisor $x_0 y_0^p+x_1 y_1^p+x_2 y_2^p=0$ in $P^2\times
P^2$, which is $SL_3$ modulo a non-reduced stabilizer containing the upper
triangular matrices. In this paper embeddings of projective homogeneous spaces
viewed as $G/H$, where $H$ is any subgroup scheme containing a Borel subgroup,
are studied. We prove that $G/H$ can be identified with the orbit of the
highest weight line in the projective space over the simple $G$-representation
$L(\lambda)$ of a certain highest weight $\lambda$. This leads to some strange
embeddings especially in characteristic $2$, where we give an example in the
$C_4$-case lying on the boundary of Hartshorne's conjecture on complete
intersections. Finally we prove that ample line bundles on $G/H$ are very
ample. This gives a counterexample to Kodaira type vanishing with a very ample
line bundle, answering an old question of Raynaud.


title: Poncelet theorems

abstract:
  If there is one polygon inscribed into some smooth conic and circumscribed
about another one, then there are infinitely many such polygons. This is
Poncelet's theorem. The aim of this note is to collect some (mostly classical)
versions of this theorem, namely:
  - Weyr's Poncelet theorem in $P_3$ (1870),
  - Emch's theorem on circular series (1901),
  - Gerbaldi's formula for the number of Poncelet pairs (1919),
  - the Money-Coutts theorem on three circles (1971),
  - the zig-zag theorem (1974),
  - a (probably new) Poncelet theorem on three conics,
  - a Poncelet formula for quadrics of revolution.


title:
On the cohomology ring of the moduli space of rank 2 vector bundles on a
  curve

abstract:
  Let $\MS_g$ be the moduli space of stable holomorphic vector bundles of rank
2 and fixed determinant of odd degree over a smooth complex projective curve of
genus $g$. This paper proves various properties of the rational cohomology ring
$H^*(\MS_g)$. It is shown that the first relation in genus $g$ between the
standard generators satisfies a recurrence relation in $g$ and that the
invariant subring for the mapping class group is a complete intersection ring.
(These two results have been obtained independently by Zagier, Baranovsky and
Siebert & Tian.) A Gr\"obner basis is found for the ideal of invariant
relations. A structural formula for $H^*(\MS_g)$ (originally conjectured by
Mumford) is verified and a natural monomial basis is given.


title:
Algebraic cohomology of the moduli space of rank 2 vector bundles on a
  curve

abstract:
  Let $\MC$ be the moduli space of stable holomorphic vector bundles of rank 2
and fixed determinant of odd degree, over a smooth projective curve $C$. This
paper identifies the algebraic cohomology ring $\HA^*(\MC)$, i.e. the subring
of the rational cohomology ring $H^*(\MC;\QQ)$ spanned by the fundamental
classes of algebraic cycles, in terms of the algebraic cohomology ring of the
Jacobian $\JC$.


title:
A Compactification Over $\overline{M}_g$ Of The Universal Moduli Space
  of Slope-Semistable Vector Bundles

abstract:
  A projective moduli space of pairs (C,E) where E is a slope- semistable
torsion free sheaf of uniform rank on a Deligne- Mumford stable curve C is
constructed via G.I.T. There is a natural SL x SL action on the relative Quot
scheme over the universal curve of the Hilbert scheme of pluricanonical, genus
g curves. The G.I.T. quotient of this product action yields a functorial,
compact solution to the moduli problem of pairs (C,E). Basic properties of the
moduli space are studied. An alternative approach to the moduli problem of
pairs has been suggested by D. Gieseker and I Morrison and completed by L.
Caporaso in the rank 1 case. It is shown the contruction given here is
isomorphic to Caporaso's compactification in the rank 1 case.


title: Koszul Property and Frobenius Splitting of Schubert Varieties

abstract:
  We show how the Frobenius splitting method of Mehta-Ramanathan implies the
Koszul property of projective coordinate rings of Schubert varieties.


title: Torsion-Free Sheaves and Moduli of Generalized Spin Curves

abstract:
  This article treats compactifications of the space of generalized spin
curves. Generalized spin curves, or $r$-spin curves, are pairs $(X,L)$ with $X$
a smooth curve and $L$ a line bundle whose r-th tensor power is isomorphic to
the canonical bundle of $X.$ These are a natural generalization of $2$-spin
curves (algebraic curves with a theta-characteristic), which have been of
interest recently, in part because of their applications to fermionic string
theory. Three different compactifications over $\Bbb{Z}[1/r],$ all using
torsion-free sheaves, are constructed. All three yield algebraic stacks, one of
which is shown to have Gorenstein singularities that can be described
explicitly, and one of which is smooth. All three compactifications generalize
constructions of Deligne and Cornalba done for the case when $r=2.$