November 7 to 9, 1997
The University of Notre Dame
Version of November 4, 1997
Chow rings of moduli spaces of pointed elliptic curves.
Pasha Belorousski
University of Chicago
Let 
be the Deligne-Mumford compactification of the
moduli space of smooth genus g curves with n marked points.
For g=0 Sean Keel (1992) determined the ring structure of the
(integral) Chow ring for all n. The other extreme - when n=0
and g;SPMgt;0 - has been extensively studied by Mumford (g=2;
1983), Faber (g=3,4; 1990), Izadi (g=5; 1994) and others. I
consider the first intermediate situation - the case when g=1
and n;SPMgt;0. More specifically:
(a) I completely determine the ring structure of the (rational) Chow ring for n=3,4 and 5 (and show that it is isomorphic to the rational cohomology ring using results of Getzler);
(b) I give some partial results for n general. These concern the structure of the so called "tautological ring", which is the numeratively significant subring of the full Chow ring.
Projections from Subvarieties.
Mauro Beltrametti, Genova, Italy
(joint project with A. Howard, M. Schneider and A.J.
Sommese.)
A classical tool for studying projective varieties is projection
from a linear subspace. To be specific, if X is an
n-dimensional subvariety of complex projective space 
, one can project to a lower dimensional 
by using a linear subspace 
as the axis projection. The usual assumption made
is that 
is generically chosen and does not lie in
X. However the case in which 
is
interesting and natural. It occurs, for example, in proving
certain vanishing theorems in cohomology. The case when k=1 was
studied by Sommese (``Hyperplane sections of projective surfaces,
I: The adjunction mapping,'' Duke Math. J. 46 (1979), 377-401)
and by Ilic (``Geometric properties of the double point
divisor,'' Trans. A.M.S., to appear) for higher dimensions. We
are interested in extending these results in two directions:
[a)] to study projection from a linear subspace of dimension greater that 1 contained in X, and [b)] to formulate and study the technique of projecting from a non-linear subspace of X.
Koszul Cohomology and k-normality of a projective
variety.
G.M. Besana,
Eastern Michigan University
Ypsilanti MI.
and A. Alzati
University of Milan, Italy
Koszul Cohomology groups give a free resolution of the ideal
sheaf of a variety X when X is projectively normal. When X
is not projectively normal Koszul cohomology only gives upper
bounds for the degree of the generators of the ring 
where L is the very ample line bundle
on X giving the embedding under consideration. If X is a
scroll over a curve or a surface or a variety fibred over a curve
in hypersurfaces of degree two and three, this is sufficient to
have information about thek-normality of X. Several results
in this direction are obtained. We show that under a weak
condition if X is a n-dimensional scroll over a curve with 
then R(X) is generated in degree 2 and thus X is
projectively normal as soon as it is 2-normal. The same is true
for varieties of dimension greater or equal to four, fibred in
hypersurfaces of degree 2 and 3 over a smooth curve. With
similar techniques, information are gathered on the possible
equations defining a very interesting family of surfaces scrolls
over a genus two curve, embedded in the five dimensional
projective space as a class of surfaces of degree eight.
The rationality of moduli spaces of vector bundles over
smooth curves.
Hans U. Boden
McMaster University
Let X be a Riemann surface of genus 
a line
bundle of degree d over X, and 
the moduli
space of semistable bundles E of rank r with determinant L.
Conjecture: is rational, i.e. it is
birational to a projective space.
Despite some positive results (mainly due to Newstead), this is
still an open problem, even for (r,d)=1. In this talk, we will
discuss the Newstead's approach to this conjecture, and explain
why it doesn't work in general. We then illustrate a method for
studying a closely related problem, namely the birational
classification of moduli spaces of parabolic bundles over X.
The result is that these moduli spaces are rational whenever one
of the multiplicities associated to the quasi-parabolic structure
is equal to one. This implies that is
stably rational, which in turn can be used to prove the
conjecture for (r,d)=1 and either (d,g)=1 or (r-d,g)=1.
Curves On Singular Surfaces In 
John Brevik
Berkeley University
I will discuss a number of examples, counterexamples, results and
conjectures relating to the general question: For a given
singular surface 
of degree d in , is it true that any
curve on is the limit of a family of curves on smooth
surfaces of degree d? Rather complete answers are given for
surfaces of degree 2 and 3, and partial results, examples,
counterexamples, and conjectures are given for surfaces of higher
degree.
Regularity and interpolation.
Marc Chardin
(joint work with Patrice Philippon)
CNRS & Université Paris VI
In this talk we explain a weak notion of regularity for a graded
module M over a noetherian positively graded algebra. Instead
of the usual condition of m-regularity we ask that, for all
i, 
where
is an homogeneous ideal, and then say that M is
-regular. This notion is especially
interesting when contains a non-zero divisor in
M, such a module is called -perfect.
Debra Coventry
Oklahoma State University
Suppose D is an effective divisor class on a smooth rational surface S and N(D) is the degree of the closure of the locus of all irreducible rational curves in |D|. Generalization of the Rational Fibration Method of Caporaso and Harris and allowing for all possible degerations of fibers gives an expression for N(D). We begin by describing the family of curves representing the set of reducible curves in |D| through a set of general points that are the limits of irreducible rational curves through these points. Allowing for all possible degenerations of fiber types and calculating the intersection pairings for the basis of these curves we are able to generate a formula for N(D).
Cohen-Macaulay Rees algebras with minimal multiplicity for
equimultiple Cohen-Macaulay ideals.
Clare D'Cruz
Tata Institute of Fundamental Research, India
Let (R,m) be a local ring of positive dimension d and let I
be any ideal of R. The Rees algebra of I is defined as
and the extended Rees
algebra of I is defined as 
where 
for 
.
Let I be an equimultiple CM ideal in R. Our objective here
is to give necessary and sufficient conditions so that the Rees
algebra R[It] localised at (m, It) and the extended Rees
algebra 
localised at 
is CM with
minimal multiplicity.
Our main results are:
Theorem Let (R,m) be a Cohen-Macaulay (CM )local
ring of dimension 
. Let I be an equimultiple CM
ideal of height h = 2. Then R[It] localised at its maximal
homogeneous ideal 
is CM with minimal
multiplicity if and only if R has minimal multiplicity, 
and 
.
Theorem Let (R,m) be a CM local ring of positive
dimension d. Let 
be an equimultiple CM
ideal of R. Suppose localised at 
is Cohen-Macaulay with minimal multiplicity.
Then 
. If 
, then 
is CM with minimal multiplicity if and only
if R has minimal multiplicity, and 
. If 
, then is CM
with minimal multiplicity if and only if R is a regular local
ring, and 
.
Singularities of divisors, vanishing theorems and
birational geometry of irregualar varieties
Lawrence Ein
University of Illinois at Chicago
We'll discuss how the vanishing theorems can be used to study the singularities of divisors on an abelian variety or a Fano manifold. We'll also disucss how the generic vanishing theorem of Green and Lazarsfeld can be used to study Albanese mapping of varieties of Kodaira dimension zero.
Geometry of the Gale Transform,
and Gorenstein sets of points
David Eisenbud
MSRI
The Gale transform is an elementary linear algebra construction
that takes a (sufficiently general) set of g points in
projective r-space 
to a set of g corresponding points in
, where g = r+s+2. It has been used in the study
geometryfrom theta functions (Coble), and recently by Sorin
Popescu and myself in the study of the minimal free resolution
conjecture for general sets of points. In this talk, following a
second paper on the subject by myself and Popescu, I will
describe a number of classical and recent geometric constructions
of the Gale transform. In particular I will focus on the case of
arithmetically Gorenstein sets of points, a topic treated by
several authors in the 19th century.
On the Depth of the Tangent Cone
Juan Elias
Departament d'Àlgebra i Geometria
Universitat de Barcelona
Gran Via 585, 08007 Barcelona, Spain
Let 
be a d-dimensional Cohen-Macaulay local ring.
In this talk we study the depth of the the associated graded ring
of R with respect a 
-primary ideal I in terms of
Vallabrega-Valla's conditions and the length of 
,
J minimal reduction of I, 
. We also study the
efectivity of the result and the growth of the Hilbert function.
The excess intersection formula and Gromov-Witten invariants
Lars Ernstrom
(report on joint work in progress with Andreas Gathmann)
Gromov-Witten invariants have in many cases been shown to compute the number of curves satisfying conditions of incidence and tangency in general position. We compute the same Gromov-Witten invariants using conditions which are not necessarily in general position. As a result the intersection often has a part of excessive dimension. We show that the contribution from the excess intersection formula is what physicists call a gravitational descendant. The part of the intersection with expected dimension is, in at least some cases, of enumerative significance, computing the number of curves satisfying, for example, conditions on the nodes and the bitangents of the curves.
Degenerations of linear systems and ramification points.
Eduardo Esteves
Instituto de Matematica Pura e Aplicada (IMPA)
Estrada Dona Castorina 110
22460-320 - Rio de Janeiro - RJ, Brazil
In the 80's, D. Eisenbud and J. Harris (Invent. math. 85) developed a general theory in order to understand what happens to a linear system and its ramification points on a smooth curve when the curve degenerates to a curve C of compact type. Eisenbud and Harris were able to obtain remarkable results from their theory, and we refer to loc. cit. for a partial list of the articles where these results are proved. In one of these articles Eisenbud and Harris asked: "What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?" (Invent. math. 87, p. 499). In this talk, I shall present an answer to this question. More precisely, I will show how to compute the limit ramification points in terms of limit linear systems. As an application, I will consider the case where C has just two components, and show how to compute the limit Weierstrass points, under certain generic conditions.
A Theorem on the Fundamental Group of the
Complement of a Union of Lines
Kwai-Man Fan
Department of Mathematics, National Chung Cheng
University
Minghsiung, Chiayi 621, Taiwan
We give in this talk a report of the results established in the
paper [1] and [2]. The main results in [1] and [2] gives a
generalization of a theorem of Zariski on the fundamental group
of the complement a union of complex projective lines in the
complex projective plane [3]. Let 
be a
union of complex projective lines. By a higher singularity of
we mean a singular point of of multiplicity
, and the term "a line" means a complex projective line in
. Let 
be the number of lines in . Let
be the number of higher singularities of .
Let 
be the set of all higher
singularities of . Let m(P) be the multiplicity of
at P and 
be the sum of
multiplicity of all higher singularities of . We need one
more number 
which is called the number of
components of in its decomposition to components in
general position.
Let 
be the union of 2 distinct
algebraic curves, we say 
intersect in nodes if
for any 
, the multiplicity of C at P equals
2. To give a definition for 
, write: 
where each 
is a union of lines. Require that: a) for 
is a subset of the set of all nodes of
,
b) for 
is not a union of nontrivial
components that intersect in nodes. That is: if
, where both
and 
are unions of lines, and
intersect in nodes, then one of
equals the empty set. We give a
proof of the existence and uniqueness of this decomposition of
is given in [2]. Denote a free group of rank a by
, and a free abelian group of rank b by 
. The main
result is the following:
Theorem. Let be a union of n
lines, and 
be the set of all higher
singularities of . Suppose
. Then
where 
.
This theorem generalizes the well known result of Zariski which
says that: if is a union of n lines in
general position, then 
.
Filtrations on cycles via Chow correspondences
Eric Friedlander
Northwestern University
We discuss a filtration on the Griffiths group which arises from a study of topological abelian groups of algebraic cycles. Using Nori's rational Lefschetz theorem for complete intersections and a relativization of techniques previously introduced for Lawson homology, we obtain examples for which this filtration has non-zero associated graded pieces.
Chern Class Formulas for Degeneracy Loci
William Fulton
University of Chicago
We will describe some formulas, proved and conjectured, for degeneracy loci involving a sequence of maps between vector bundles with rank conditions on arbitrary composites. This is joint work with Anders Buch and Sergey Fomin.
Catalecticant Varieties
Anthony Geramita
Queen's University Kingston, Ontario (Canada)
Universita' di Genova, Genova, Italia
Catalecticant matrices generalize, in a natural way, both generic symmetric matrices and classical Hankel matrices and catalecticant varieties are the projective schemes defined by the ideals of minors of catalecticant matrices.
The ideals of minors of Hankel matrices are the defining ideals of certain secant varieties to rational normal curves and are fairly well understood. In contrast, the schemes defined by the minors of catalecticant matrices offer many new and interesting problems.
In this lecture I will discuss these classical results and explain some of the new problems that have arisen from the catalecticant matrices.
The fact that there are connections between these studies and questions about the structure of artinian Gorenstein rings was first made clear through the work of A. Iarrobino and V. Kanev. I will explains some of these connections also and show how to use them to gain some insight into the nature of catalecticant varieties.
Complete Kahler Metrics for Singular Algebraic Varieties.
Caroline Grant
We describe natural constructions of complete Kahler metrics on the nonsingular set of an algebraic variety, or more generally, on a subvariety of a compact Kahler manifold. The constructions use explicitly the geometry of a sequence of blow-ups which resolve the singularities of the variety. In the case of a variety with isolated singularities, our metrics have the same order of growth as Saper's metrics, whose L2-cohomology he proved is isomorphic to the intersection cohomology of the variety. The results discussed represent joint work with P. Milman.
Quadruple Covers of Algebraic Varieties
David W. Hahn (with R. Miranda)
Let X and Y be varieties over a field 
is a quadruple cover of Y if 
is a locally free, rank 4 
-algebra. If char
, we see that splits as 
where 
is a locally free rank 3
sheaf over , which is locally the `trace zero'
module. For each 
, we therefore have a rank 4
associative, commutative algebra over 
. We find
that these algebras are parametrized by an affine cone over the
Grassmanian G(2,6) with vertex corresponding to the algebra
. We then show that a quadruple cover with
trace zero module over a variety Y is determined by
a totally decomposable section
.
We then examine the case in which the section 
has no
zeros. Here, each rank 4 algebra may be associated to a pencil
of conics. As a special case of this, we look at the work of
G. Casnati on Gorenstein covers, and we show that his analysis is
the subcase where the pencil of conics has length 4 base locus.
Finally, we study the case in which the trace zero module is split. In this context, Galois covers, which are covers
induced by the action of a group of order 4 on the covering
variety X, are also studied.
Cubics
Joseph Harris
Harvard University
The talk is about recent work of Brendan Hassett (and earlier work of Beauville-Donagi and Voisin) on cubic fourfolds. The general question is which smooth cubic fourfolds are rational and which (if any) are not; it's very much open at this point. The intriguing thing is that there is some reason to believe that the locus of smooth cubic fourfolds that are rational is neither open nor closed in the moduli space of smooth cubic fourfolds-that it may be a countable, dense union of subvarieties of the moduli space. The reason for this belief has to do with the Hodge structures of cubic fourfolds, which bears an interesting relation with those of K3 surfaces.
Limiting Plane Curves and the Minimal Model Program
Brendan Hassett
University of Chicago
Several years ago, Kollár and Alexeev constructed geometric
compactifications of moduli spaces of surfaces of general type.
In these compactifications, computing the stable reduction of a
one-parameter family of surfaces boils down to finding the
canonical model of the total space of the family (at least after
a suitable base change). This work generalizes to log surfaces of
general type, i.e. pairs (C,S) where C is a smooth curve
imbedded in a smooth surface S so that 
is big.
Perhaps the simplest example of such pairs are curves in the
projective plane with degree at least four. The boundary points
for the corresponding moduli spaces are `limiting plane curves'.
They consist of a stable curve C contained in a surface S
that is a singular degeneration of some Veronese imbedding of the
plane. The singularities of (S,C) must satisfy certain
restrictions. A complete classification of such surfaces would
yield a description of the closure of the plane curve locus in
the moduli space of stable curves 
. We give a
partial classification of these singular surfaces and the
corresponding curves. We emphasize the cases where the limiting
curves are smooth, corresponding to points of the closure of the
plane curve locus in the moduli space 
.
Boundary manifolds of algebraic plane curves complements
Eriko Hironaka
University of Toronto
In this talk we will apply techniques from the theory of graphs of manifolds and graphs of groups to study the boundary of a regular neighborhood of an algebraic curve in the complex projective plane. This boundary manifold can be realized as a graph of well understood 3-manifolds connected along tori and its fundamental group is a corresponding graph of groups. We will discuss how this graph of groups relates to the fundamental group of the algebraic plane curve complement and give some consequences.
Problems in the theory of tight closure
Craig Huneke
Purdue University
This talk will be a survey of tight closure. Tight closure was
introduced in 1986 by Melvin Hochster and myself. It is a closure
operation on ideals in equicharacteristic Noetherian rings. If
R is such a ring, and I an ideal, the tight closure of I is
denoted 
.
Originally, the impetus for defining tight closure came from three sources: work on the homological conjectures in commutative algebra, work on the Cohen-Macaulay property of rings of invariants and understanding the concept of F-purity, and finally work on the integral closure of ideals, especially the so-called Briançon-Skoda Theorem. Since this time, tight closure has proved a fundamental tool in the proofs of different theorems: the existence of Big Cohen-Macaulay algebras, the proof of a uniform Artin-Rees theorem, and more recently in a more algebraic understanding of the Kodaira vanishing theorem and `Frobenius' characterizations of singularities paralleling the breakdown into terminal, canonical, log-terminal, and log-canonical singularities in algebraic geometry. It has signaled a renewed interest in the properties of the Hilbert-Kunz function.
The definition of tight closure is by reduction to positive
characteristic and in some ways tight closure simply codifies the
method of reduction to prime characteristic. However, making the
definition leads to an understanding of certain invariants of
rings such as the `test ideal', defined to be the ideal given by
the intersection of 
, where the intersection runs over all
ideals of R.
The talk will focus on results, open problems and recent progress in this subject.
Gorenstein divisors on ACM codimension two subschemes of 
A. Iarrobino (with V. Kanev)
Northeastern University
When does a graded height 3 graded Gorenstein ideal I of 
contain a unique graded arithmetically
Cohen-Macaulay (ACM) ideal J of height 2, such that W =
Spec(R/I) is a "tight divisor" on Z =Spec(R/J)? Here W is
tight if the multiplicity of Z is the maximum value of the
n-3 difference 
of the Hilbert
function H(R/I). The following theorem partially answers a
question posed by A. Geramita and J. Migliore in [GM].
Theorem. (joint with Vassil Kanev [IK]). If I is a
Gorenstein height 3 ideal of R, and contains a subsequence (s,s,s) and T
has socle degree 
, then 
defines an ACM height two ideal of R, having multiplicity
s, such that W = Spec(R/I) is a "tight divisor" on Z =
Spec(R/J). Furthermore, Z is the unique subscheme of on which W is a tight divisor.
The Jacobian Conjecture for 
as a problem about
divisors
on rational surfaces, and graphs associated to them
David B. Jaffe
(report on joint work in progress with A. J. Radcliffe)
[Abstract for talk at Midwest Algebraic Geometry
Conference, Notre Dame, 1997]
Given a morphism 
over
, we can find a blowup 
and a map 
which extends
f. Let L be the line at infinity in 
. By
construction the divisor 
is a linear combination
of exceptional curves and the strict transform of the line at
infinity on the ``other'' . Now suppose that f is
étale. (Then the Jacobian Conjecture is equivalent to the
assertion 
.) By a careful consideration of
ramification, we obtain restrictions on D. Moreover, a
consideration of the combinatorial data associated to 
and
D suggest that by finding additional restrictions on this data
(arising from new geometric insight), one may be able to settle
the Jacobian Conjecture by combinatorial arguments.
Intersection numbers and rank one cohomological field
theories in genus one.
Alexandre Kabanov
(This is a joint work with T. Kimura)
We obtain a simple, recursive presentation of the tautological classes on the moduli space of curves in genus 0 and 1 in terms of the boundary strata. We derive differential equations for the generating functions for their intersection numbers. As an application, we describe the moduli space of normalized, rank one even cohomological field theories in coordinates which are additive under taking tensor products. Our results generalize those of Kaufmann, Manin, and Zagier.
Galois coverings of the projective line with group
and principally polarized abelian varieties of
dimension
six.
Vassil Kanev
A classical theorem of Wirtinger states that the general principally polarized abelian variety of dimension 5 is a Prym variety of a double unramified covering of a curve of genus 6. This fact was used by R. Donagi in his proof of the unirationality of A5. The result I want to lecture on is that the general principally polarized abelian variety of dimension 6 is isomorphic to a Prym-Tjurin variety associated to a covering of the projective line of degree 27, with monodromy group equal to the Weyl group of type E6 acting on the 27 lines on a cubic surface, with 24 branch points, where the monodromy at each branch point is the simplest possible, i.e associated with a reflection.
Weak semistable reduction in characteristic 0.
Kalle Karu
In this talk I will describe our recent work on extending the
semistable reduction theorem of a morphism of varieties 
in characteristic 0 to a base of arbitrary dimension. We reduce
the varieties to toroidal embeddings, and state the problem in
terms of the combinatorics of the associated polyhedral
complexes. We have been able to prove semistable reduction for
morphism of small relative dimension and a slightly weaker form
of semistable reduction in the general case.
Quantum Contact Cohomology of the Projective Plane.
Gary Kennedy
Ohio State
This is joint work with Lars Ernström. We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane, which is itself a deformation of the usual cohomology ring. Just as the quantum cohomology encodes the incidence characteristic numbers of rational plane curves, the quantum contact cohomology encodes the tangency characteristic numbers.
By November I may also be able to report on more recent work with Ernstrom and Susan Colley, and I may wish to update the title to ``Higher order quantum cohomology of the projective plane.''
Curves in 
-trivial Threefolds
Holger Kley (joint work with Herb Clemens)
University of Utah
We study the deformation theory of curves of arbitray genus in K-trivial threefolds (which may have isolated singularities). We relate the obstruction sheaf to the sheaf of differentials on the Hilbert scheme and deduce an enumerative formula. The main application is to the existence and counting of curves on Calabi-Yau complete intersections: starting with a K3 surface containing smooth curves, we build a nodal threefold and show when the curves break up to isolated curves under a general deformation to a smooth threefold. This significantly extends results of Clemens and S. Katz for g=0 and Kley for g=1.
A preprint will be available at the time of the conference.
On invariants of local rings.
Jee Koh
Dept. for Math., Indiana University, Bloomington, IN 47405
Kisuk Lee
Global Analysis Res. Center Seoul National Univ., Seoul,
Korea
We define several invariants of (generalized) local rings arising from certain restrictions on infinite minimal free resolutions. We show that the (Auslander) index of a Gorenstein local ring and the (Castelnuovo-Mumford) regularity of a Cohen-Macaulay homogeneous ring agree with some of these invariants. We show that Ding's conjecture holds for a Gorenstein local ring satisfying a certain condition which seems weaker than requiring the associate graded ring be Cohen-Macaulay (which was assumed by Ding).
A cohomological surjectivity theorem for rational and
(some) non-rational singularities.
Sandor Kovacs
Department of Mathematics
Massachusetts Institute of Technology
The next theorem represents a culmination of the work of the following authors: Tankeev, Ramanujam, Miyaoka, Kawamata, Viehweg, Kollár, Esnault-Viehweg.
Theorem.Kollár95, 9.12. Let X be a proper variety
and L a line bundle on X. Let 
,
where 
is an effective divisor and let s be a
global section whose zero divisor is D. Assume that 
for every i. Let Z be the normalization of taking the
root of s. Assume further that
is surjective. Then for any 
the natural map
is surjective. This theorem has a wide range of applications. Most notably it is used in the proof of the most generic versions of the Kodaira vanishing theorem.
If Z is a smooth proper variety, then Hodge theory implies that
is surjective. In
the applications however, Z will not be always smooth, so it is
interesting to know what singularities one can allow on Z to
still have that surjectivity.
Steenbrink's conjecture states that if Z has rational
singularities, then 
is surjective. (In fact his
conjecture predicts a more precise local version.) Steenbrink
himself proved the local variant of the conjecture for isolated
rational singularities. Kollár proved the conjecture if Z is
projective and conjectured that log canonical singularities have
the same property (cf. Steenbrink83, Kollár95, Ch.
12, Kollár et al. 92, 1.13). Kollár's conjecture holds
for isolated Gorenstein singularities by Ishii85.
Here we prove Steenbrink's (local) conjecture and Kollár's conjecture under some extra conditions:
Theorem Let Z be a complex variety with either rational
or log canonical 
-Gorenstein singularities. Let
be the set of singular points of Z, and
let 
denote the smallest closed subset of Z, such
that 
has rational singularities. Assume that
.
Then the Kollár-Steenbrink conjecture holds. In particular if Z is proper, then
is surjective.
Vector bundles on Projective spaces in positive
characteristic
N. Mohan Kumar
Department of Mathematics
Washington University
St. Louis, MO 63130
We prove a criterion for the existence of a vector bundle on
projective n-space by the existence of certain vector bundles
in (n-1)-space. Though this criterion has nothing to do with
the characteristic, we use this criterion to construct many rank
two vector bundles on 
over fields of any positive
characteristic. The construction yields both stable and unstable
indecomposable bundles and we compute their chern classes. We
will also discuss some appropriate `universal space' for
constructing these bundles. This work is expected to appear soon
in print.
Castelnuovo-Mumford Regularity of Smooth Projective
Varieties of dimension above two.
Sijong Kwak
Korea Institute for Advanced Study
In this talk, I deal with the well-known problem, i.e., Castelnuovo-Mumford regularity.
Generic projection method has been useful in solving such a
problem in the cases of curves and surfaces. To apply this
method to the higher dimensional smooth varieties of dimension
above two, we have to control the fibers of a generic projection
in order to separate finite schemes appearing in the fibres of a
generic projection by polynomials of fixed degree. (i.e., we
have to deal with k-normality of finite scheme appearing in the
fibers of generic linear projection.) Roughly speaking, this can
be done by using J. Mather's Theorem on linear stable projections
and stratification according to the length of fibers under the
generic linear projection and as a result, we succeed in getting
more nice bounds at least in the cases of dimension less than 6,
namely, reg(X) is less than or equal to 
. This type
of material will be appearing in the journal of algebraic
geometry.
Remarks on the pluricanonical linear series on threefolds
Seunghun Lee
In their paper ``Global generation of pluricanonical and adjoint
linear series on smooth threefolds", L. Ein R. Lazarsfeld showed
that, among other things, 
is base point free for a
smooth projective threefold X if 
. Here we deal with
the rest of the cases, i.e. we show that for a Gorenstein
projective minimal threefolds X of general type, 
is
base point free and 
is also base point free if 
.
We also show that separate two distinct points if 
.
REES ALGEBRAS OF FINITELY GENERATED TORSION-FREE MODULES
OVER A
TWO-DIMENSIONAL REGULAR LOCAL RING
Jung-Chen Liu
Purdue University
Let be a two-dimensional regular local ring and let
A be a finitely generated torsion-free R-module. If A is a
complete module, then Katz and Kodiyalam show A satisfies five
conditions, one of these being that the Rees algebra 
of A is Cohen-Macaulay and another being that the ``associated
graded ring'' 
of A is Cohen-Macaulay.
They ask whether these five conditions are equivalent without
assuming A to be complete. We exhibit an example to show that
may be Cohen-Macaulay while
fails to be Cohen-Macaulay, and investigate other implications
among these five properties in the case where A is not
complete. We prove in general that the depth of is
greater than or equal to the depth of .
Ample vector bundles of small curve genera.
H. Maeda
Let 
be a vector bundle of rank n-1 on a smooth
complex projective variety X of dimension 
, and let
be the curve genus of 
defined
by the formula 
, where 
is the canonical bundle of
X. Then it is proved that is a nonnegative
integer if is ample. Moreover, polarized pairs
with 
are completely
classified.
Kawachi's invariant for normal surface singularities.
Vladimir Masek
Washington Univ. in St. Louis
We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic Gorenstein or rational (without assuming a priori that they are Q-Gorenstein).
Next, we discuss effective results (stated in terms of Kawachi's invariant) regarding global generation of adjoint linear systems on normal surfaces with boundary. Such results can be used in proving effective estimates for global generation on singular threefolds. The theorem of Ein-Lazarsfeld and Kawamata, which says that the minimal center of log-canonical singularities is always normal, explains why the results proved here are relevant in that situation.
Cohomology of complete intersections in toric varieties
Anvar Mavlyutov
We explicitly describe Hodge structures of complete intersections in compact simplicial toric varieties.
Let 
be a complete simplicial
d-dimensional toric variety and let 
be its homogeneous ring with variables
corresponding to the 1-dimensional cones in the
fan . This ring has a natural grading by the Chow group
.
A closed subset 
defined by homogeneous polynomials 
is called a
quasi-smooth intersection if X is a V-submanifold
(suborbifold) of 
and the codimension of X in
is s. This notion generalizes a nonsingular
complete intersection in a projective space.
A Lefschetz-type theorem. If is an intersection of ample divisors,
then the natural map 
is an isomorphism for i;SPMlt;d-s and an injection for
i=d-s.
The variable cohomology group 
is the
For 
the
Jacobian ring R(F) denotes the quotient of R by the ideal
generated by the partial derivatives 
, 
, 
,
. If 
with homogeneous
, this ring carries a natural grading by the
group 
, so that
, 
,
.
Theorem. Let be a d-dimensional
complete simplicial toric variety, and let 
be a quasi-smooth intersection of ample divisors defined by
, . If
, then for 
, we
have a canonical isomorphism
where 
,
. In the case
there is an exact sequence
where 
is the cohomology class of
X.
Intermediate rings 
and the residual field structure of their normalizations.
Mark McCormick
Let A be an excellent reduced local ring with Henselization
and strict Henselization 
. Then the splitting
of minimal prime ideals across the extension 
is determined by the residual field
structure of the normalization of A. For a local intermediate ring dominated by
, D is etale over A if and only if the normalization
of D is semilocal and has an appropriate residual field
structure. Furthermore, if A is Henselian there is a similar
characterization for the Noetherianness of D.
Vector nonlinear Schroedinger hierarchies as approximate
Kadomtsev-Petviashvili hierarchies
Peter Miller
Institute for Advanced Studies, Princeton
The Kadomtsev-Petviashvili (KP) hierarchy, a collection of compatible nonlinear equations, each in 2+1 independent variables, can be consistently constrained in many different ways to yield hierarchies of equations in 1+1 independent variables. In particular, the N-component vector nonlinear Schroedinger (VNLS) hierarchies are contained within the KP hierarchy in this way. These hierarchies approximate the KP hierarchy in the limit of large N, and this permits the equations of the KP hierarchy to be approximated by nonlinear equations in 1+1 dimensions.
Sharp Examples of Some Bounds on Castelnuovo-Mumford
Regularity
Chikashi Miyazaki
Nagano National College of Technology
Let X be a projective scheme of 
over a field K.
Let 
be the polynomial ring and 
be the irrelevant ideal. Then we put 
. We denote by 
the ideal
sheaf of X. Then X is said to be m-regular if 
for all 
. The
Castelnuovo-Mumford regularity of 
is
the least such integer m and is denoted by reg (X). The
interest in this concept stems partly from the well-known fact
that X is m-regular if and only if for every 
the
minimal generators of the p-th syzygy module of the defining
ideal I of occur in degree 
.
Let k be a nonnegative integer. Then X is called k-Buchsbaum if the graded S-module
is annihilated
by 
for 
, see, e.g., MM. Further
we call the minimal nonnegative integer n, if there exists,
such that X is n-Buchsbaum, as the Migliore-Miró Roig
number of X and denote by k(X). In case X is not
k-Buchsbaum for all 
, we put 
. Note
that 
if and only if X is locally Cohen-Macaulay
and equi-dimensional.
In recent years upper bounds on the Castelnuovo-Mumford
regularity of a projective variety X have been given by several
authors in terms of 
, 
, 
and
k(X). The following bound, firstly obtained in NS, is
the most optimal among the known results. Even so, whether such
bound is sharp has been still a question.
Gorenstein liaison and determinantal schemes
Uwe Nagel
Universität-GH Paderborn
We report on joint work with J. Kleppe R. M. Miro-Roig, J. Migliore and C. Peterson.
Liaison theory has mainly centered around linkage by complete
intersections. It is well understood for subschemes of
codimension two but much less is known in higher codimension. In
this talk we consider the coarser equivalence relation which is
generated by linkage using arithmetically Gorenstein subschemes
and focus on subschemes of codimension 
of projective
space. We will indicate that many results in codimension two have
analogues in higher codimension using Gorenstein liaison.
One of the starting points of liaison theory is Gaeta's result
that every arithmetically Cohen-Macaulay subscheme of codimension
two is in the liaison class of a complete intersection (licci).
It is known that the corresponding result does not hold in higher
codimension using linkage by complete intersections. Contrary to
that we conjecture that every arithmetically Cohen-Macaulay
subscheme of codimension three is in the Gorenstein liaison class
of a complete intersection (glicci). We obtain evidence for this
hope by considering standard determinantal subschemes. A
projective subscheme of codimension c is said to be standard
determinantal if its homogeneous ideal is generated by the
maximal minors of a homogeneous 
matrix. The
main result, which we want to present, says that every standard
determinantal subscheme is glicci. This is a full generalization
of Gaeta's result since it is known for subschemes of codimension
two that arithmetically Cohen-Macaulay subschemes are standard
determinantal and that arithmetically Gorenstein subschemes are
complete intersections. Note that there are standard
determinantal schemes in any codimension which are not
licci.
Decoding of One-Point Codes
Michael E. O'Sullivan
Univ of Puerto Rico
The most widely studied error-correcting codes from algebraic
geometry are ``one-point codes.'' These are constructed by
choosing rational points Q and 
on a smooth
curve X and evaluating functions on X having poles only at
Q, at the points . The image of L(mQ) in
is a the decoding code. When the curve
is 
, the construction gives Reed-Solomon codes,
which are widely used in commercial applications today. I will
show how the Berlekamp-Massey algorithm and the key equation,
which were designed for decoding Reed-Solomon codes, can be
extended to one-point codes. The algorithm uses the ``syndrome''
to find a Gröbner basis for the ideal of a certain subset of
the 
. Some interesting geometry comes into play: Using
Serre duality, the syndrome may be interpreted as a differential
adele, which provides for easy computation of error values. In
simulations with a Hermitian curve the algorithm performed much
better than expected. The explanation is that general sets of
points are easier to correct. This motivates an arithmetic
problem: what proportion of the subsets of t points among
are general?
Localization and GW-invariants
Rahul Pandharipande
A localization formula for torus equivariant perfect obstruction theories will be explained. The main application to Gromov-Witten invariants (and other related integrals) will be discussed. This talk represents joint work with T. Graber.
Associated graded rings of m-primary ideals
Claudia Polini
Michigan State University
We study the depth of the associated graded ring of an
m-primary ideal I of a Cohen-Macaulay local ring (R, m) in
terms of certain invariants of I. Precisely, given a minimal
reduction J of I we look at m-primary ideals for which
for some n and 
for k=1, ..., n-1. We show that the associated
graded rings of such ideals are ``almost'' Cohen-Macaulay.
Furthermore we give sufficient conditions on the ideal structure
for when the associated graded ring is actually Cohen-Macaulay.
Projective normality and Syzygies of Algebraic surfaces
and Calabi-Yau threefolds.
B.P.Purnaprajna
Okalhoma State University
To study equations of algebraic varieties embedded in projective space has been one of the central questions in algebraic geometry. Recently Green and Lazarsfeld extended the study of equations to the study of free resolutions of the homogeneous coordinate ring of the variety in projective space. We will talk about our recent results for algebraic surfaces and Calabi-Yau threefolds. We give here an outline of the results: We develop techniques to compute Koszul cohomology groups for several classes of carieties. As an application of this we prove results on projective normality and syzygies for algebraic surfaces and Calabi-Yau threefolds. From more general results we obtain in particular the following:
Rank two bundles on P3.
A. Prabhakar Rao
University of Missouri-St. Louis
On , we show that mathematical instantons in characteristic
two are unobstructed. We produce upper bounds for the dimension
of the moduli space of stable rank two bundles on in
characteristic two. In cases where there is a phenomenon of good
reduction modulo two, these give similar results in charcteristic
zero. We also give an example of a non-0reduced component of the
moduli space in characteristic two.
A group action on a quotient scheme and output feedback
equivalence.
M. S. Ravi
East Carolina University
We will show that the concept of output feedback equivalence in Systems theory, is related to the action of PGL(m+p) on the space of maps from the projective line to Grass(m,m+p). We will then discuss properties of the quotient space, in the sense of geometric invariant theory.
1-Semiquasihomogeneous Singularities of Hypersurfaces in
positive Characteristic
Marko Roczen
In arbitrary characteristic different from 2, the singularities
with semiquasihomogeneous equations, characterized by the
condition to have Saito-invariant 1 are the "classical"
quasihomogeneous ones, known over the field of complex numbers as
simple elliptic singularities (found by K. Saito). This is
different in characteristic 2: In odd dimensions and for weights
and 
non-quasihomogeneous equations
appear.
Also, the singularity gives an infinite family of
nonisomorphic singularities with fixed principal part, contrary
to the classical case of simple elliptic singularities, which
have modality 1 only (coming from the absolute invariant in the
principal part). The proof uses the computer algebra system
SINGULAR.
AMS classification: 14B05, 14B12, keywords: semiquasihomogeneous singularities, base field of positive characteristic
Affine Elimination Theory
J. Maurice Rojas
(MIT)
We propose a new resultant operator, the affine sparse resultant, giving a new technique for computing projections of affine varieties. In particular, this extends the applicability of the sparse resultant, which has already proved an efficient method for dealing with certain subvarieties of the algebraic torus. We thus obtain the following two corollaries: (1) A refinement of the classical Chow form, and (2) a new explicit formula for the degree of an intersection of n algebraic hypersurfaces in affine n-space. The latter formula provides a definitive generalization of Bernshtein's mixed volume formula, including all its prior extensions to affine space.
A new look at the singularities in Mori's Program.
Karen Smith
University of Michigan/M.I.T.
This talk will compare some of the singularities arising in the minimal model program to the singularities that have arisen in tight closure theory. Recent comparison theorems showing, for example, that log-terminal singularities are equivalent to F-regular singularities, will be overviewed. Results and speculations comparing certain important ideals arising in each theory (for example, the multiplier ideal and the test ideal) will be discussed. It will not be assumed that the audience is expert in Mori's program or in tight closure.
Real enumerative geometry and Shapiro's conjecture
Frank Sottile
University of Toronto
While it is difficult to obtain general results in real algebraic geometry, applications demand that we try. The study of enumerative geometry over the reals is one area motivated by applications, such as the control of linear systems. Recent work suggest that it is fruitful to study when a given enumerative problem may be solved completely over the reals. By this, we mean that there exist real conditions such that all of the (a priori complex) solutions are in fact real. A recent conjecture of Boris Shapiro and Michael Shapiro concerning the Schubert calculus of enumerative geometry on a general flag manifold gives a precise method to select real Schubert varieties all of whose (finitely many) points of intersection are real. A positive resolution of this conjecture would have direct applications to control theory. Also, the conjecture suggests a link between real enumerative geometry and Lusztig's theory of total positivity.
In this talk, we will first describe some results in real enumerative geometry and then state Shapiros' conjecture. Next, we will indicate its connection to the control of linear systems, discuss the computational evidence for it, and show its relation to total positivity.
Intersection numbers of the moduli spaces of flat
connections over Riemann surfaces.
Andras Szenes
(MIT)
We describe the structure of intersection numbers of moduli spaces of parabolic bundles over a smooth curve using iterated residues.
William Traves
Toronto
The notion of D-simplicity is used to give a new and very short proof of Nakai's conjecture for curves defined over a field of arbitrary characteristic.
Ravi Vakil
Harvard University
Classical degeneration methods can be used to calculate top intersection numbers of classes on components of moduli spaces of stable maps. We discuss applications in (i) calculating characteristic numbers of rational and elliptic curves in projective space, (ii) counting curves on rational ruled surfaces, and (iii) computing genus g Gromov-Witten invariants of Fano surfaces.
Syzygies, Secant Varieties, and Flips
Peter Vermeire
Univ of North Carolina at Chapel Hill
Let X in 
be a smooth projective variety,
scheme-theoretically defined by quadrics 
and
assume that the trivial syzygies (i.e. 
are
generated by linear ones; in particular, one can assume Green's
condition 
, i.e. X is projectively normal,
ideal-theoretically defined by quadrics, and all of the syzygies
are generated by linear ones. The 
then define a rational
map to which one can resolve to a morphism T by blowing
up along X. Excluding a small class of varieties, we show that
the map T is an embedding off of the proper transform of the
secant variety. With the hypotheses that X does not contain a
line or a plane quadric, we show that T restricted to the
proper transform of the secant variety is a 
-bundle over
, the Hilbert scheme of length 2 subshemes of X,
embedded in . From this we derive many immediate
consequences, such as properties of smoothness, normality, and
non-deficiency of the secant variety and its proper transform. We
will discuss extensions of these ideas to higher secant
varieties, as well as attempts to construct examples of flips
motivated by Michael Thaddeus' work on changes in the
linearization of GIT quotients. We also obtain a quick, and
slight, extension of a vanishing theorem of Bertram, Ein, and
Lazarsfeld for twisted powers of ideal sheaves.
Subintegral Closure of Ideals and Weak Normalization of
Rees Rings.
Marie A. Vitulli
University of Oregon
M.A. Vitulli and J.V. Leahy recently introduced the notion of the weak subintegral closure of an ideal of a commutative ring A in an extension ring B. This gives rise to an ideal in the weak normalization of A in B and enables one to identify the homogeneous components of the weak normalization of the Rees ring A[It] in B[t]. In this talk the author will discuss recent developments involving the weak subintegral closure of ideals.
On General Inverse Eigenvalue Problems
Alex Wang
Texas Tech University
Any additive or mutiplicative inverse eigenvalue problem with perturbation from a linear space L can be considered as a intersection problem of a projective variety in the Grassmanian of n-subspaces in 2n-space. In this talk we will study such varieties. We will give the defining equations of the varieties and compute the generic degrees.
Moduli of numerical Godeaux surfaces
Caryn Werner
University of Michigan
A numerical Godeaux surface is a minimal surface of general type
with zero geometric genus and 
. It is known that the
torsion group of these surfaces is cyclic of order less than or
equal to 5. While the moduli space of the surfaces with torsion
of order 3, 4, and 5 have each been shown to be irreducible of
dimension 8, the problem of determining the moduli of numerical
Godeaux surfaces with trivial and order 2 torsion are still open.
Examples of these surfaces with trivial fundamental group have
been of particular interest, as they provide examples of surfaces
homeomorphic but not diffeomorphic to an eight-fold blowup of the
plane.
Recently P. Craighero and R. Gattazzo gave an example of a
numerical Godeaux surface as the minimal desingularization of a
quintic surface in . In joint work with Igor Dolgachev, we
show that their example is simply connected. The only previous
example of a simply connected numerical Godeaux surface is due to
R. Barlow; by construction, her examples contain four
(-2)-curves. However we prove that the Craighero-Gattazzo
surfaces have no (-2)-curves, thus providing the first example of
a simply connected surface with vanishing geometric genus and
ample canonical class.
This talk will outline two different methods for constructing
these surfaces, as a singular quintic in and a realization
as a double cover of the plane. We then show how to use certain
hyperelliptic fibrations on the surface to determine the
fundamental group. The talk will conclude with a plan for using
these fibrations to distinguish among components of the moduli
space of numerical Godeaux surfaces.
Sheaf Cohomology and Linear System Theory
Bostwick Wyman
Ohio State University
Joint Work with Vakhtang Lomadze, Georgian Academy of
Sciences, Tbilisi, Georgia
Cohomology of coherent sheaves on the projective line gives an elegant and useful approach to linear system theory. In this talk we discuss controllability and observability for singular linear systems and the fundamental pole-zero exact sequence for these systems.
Castelnuovo's Lemma and h-vectors of Cohen-Macaulay
Homogeneous Domains
Kohji Yanagawa
Department of Mathematics, Graduate School of Science,
Osaka University, Toyonaka, Osaka 560, Japan
If A is a Cohen-Macaulay homogeneous algebra over a field k,
there are positive integers 
satisfying
where d is the
Krull dimension of A. We call the vector 
the h-vector of A.
Let A be a Cohen-Macaulay homogeneous integral domain
over an algebraically closed field k of 
with the h-vector 
. It is well known
that 
for all 
. We will show
that if the equality 
holds for some 
, then 
(when 
, the condition 
also implies the same
assertion). Even if 
, this statement
remains true when A is also normal. To prove this result, we
will modify Castelnuovo's argument in his study on curves of
maximal genus.
On the Lojasiewicz Exponent at Infinity for
Polynomial Functions
Alexandru Zaharia (with Laurentiu Paunescu)
For a polynomial 
with only isolated
singularities, the 
ojasiewicz number at infinity, 
, is
The following Theorem is a reformulation of a result in [CK11]:
[Theorem.]Let 
be a polynomial function. Then
there exists a polynomial automorphism 
if and only if g has no critical values and
.
We show that this Theorem can not be extended to the case of a
polynomial function , when 
.
Namely, for every 
we construct a
polynomial automorphism 
such that 
. Moreover, our polynomials 
show that several
classes of polynomials with ``good'' behavior at infinity,
considered in [Ne1], [NZ1], [Pa1] and [ST], are distinct.
ojasiewicz à l'infini pour les
applications polynomiales de 
dans et les
composantes des automorphismes polynomiaux de , C.
R. Acad. Sci. Paris, Série I, 315, 1992, 1399-1402.
Characterize uniquely K3 surfaces in terms of their
automorphisms and fixed loci
D. -Q. Zhang (with K. Oguiso)
National Univ. of Singapore
We will report our recent works with K. Oguiso. We will also prove:
Theorem 1. Let p be any of three numbers 13, 17 and 19. Then there is a unique K3 surface X with an automorphism g of order p. Moreover the pair (X,;SPMlt;g;SPMgt;) is also unique up to equivariant isomorphisms.
Remark 2. (1) By a result of Nikulin, if g is an automorphism
on a K3 surface, then the value of Euler's 
-function at
the point ord(g) is less than or equal to 21. In particular,
if the order ord(g) equals a prime number p, then p is less
than or equal to 19.
(2) Theorem 1 is not true for p less than or equal to 11. However, we have similar uniqueness theorem for all p, like Theorem 3 below, in terms of the number of g-fixed curves.
Theorem 3. Let X be a K3 surface with an automorphism g of order m equal to 2 (resp. 3). Suppose that g acts non-trivially on 2-forms, there is no g-fixed (point-wise) curves of genus at least 2, and there are at least 10 (resp. 6) g-fixed rational curves. Then such a pair (X, ;SPMlt;g;SPMgt;) is unique up to isomorphisms. To be precise, X is the unique K3 surface of Picard number 20 and discriminant 4 (resp. 3).
Remark 4. Theorem 3 is one of the results in the paper "The most algebraic K3 surfaces and the most extremal log Enriques surfaces, by K. Oguiso and D. - Q. Zhang" accepted for publication by Amer. J. Math.