Commutative Algebra:
Geometric, Homological, Combinatorial and Computational Aspects
Sevilla, Spain
June 1821, 2003

Some invariants of local rings
by Josep Àlvarez (Universitat Politècnica de
Catalunya, Spain)
Abstract:
Let R be a formal power series ring over a field of characteristic
zero and I Í
R be any ideal. The aim of this work is to
introduce some numerical invariants of the local rings R/I by
using theory of algebraic Dmodules. More precisely, we will
prove that the multiplicities of the characteristic cycle of the
local cohomology modules H_{I}^{ni}(R)
and H_{P}^{p}(H_{I}^{ni}(R)), where
P subseteq R is any prime ideal that contains I,
are invariants of R/I.

Saturation and CastelnuovoMumford regularity
by Isabel Bermejo (Universidad de la Laguna, Spain)
Abstract:
Let R be a polynomial ring in n+1 variables over an
infinite field k, and let I be a homogeneous
ideal of R. By avoiding the construction of a minimal graded free
resolution of I, we provide an effective method for computing the
CastelnuovoMumford regularity of I.
As an application, we obtain an explicit formula for the
CastelnuovoMumford regularity of the defining ideal of a projective monomial
subvariety X of P^{n}_{k} of codimension two.
All the results have been implemented
in Singular and will be included in the
new version of our joint library
with G.M. Greuel, mregular.lib.
This is joint work with Philippe Gimenez (Universidad de Valladolid).

Multiplier ideals in exotic settings
by Ana Bravo (Universidad Autónoma de Madrid, Spain)
Abstract:
We describe different approaches to develop a notion of multiplier
ideal in the context of nonsmooth varieties, and over fields of
arbitrary characteristic. This is joint work with Karen Smith
(University of Michigan).

Asymptotic Behaviour of Cohomology
by Markus Brodmann (Universität Zürich, Switzerland)
Abstract:
We present a few results on the asymptotic behaviour for n « 0 of the
cohomology modules of the nth twist F(n) of a coherent sheaf F on a
projective scheme X (over an affine noetherian base scheme Y).
If Y is either of dimension 1 or else semilocal and of dimension 2, the set of
asymptotic primes of the mentioned cohomology modules eventually stabilizes
for n « 0 (under mild additional hypotheses on Y).
If Y is local and of dimension 1, various numerical invariants (multiplicities,
Hilbert coefficients, number of generators, socle dimensions, torsion lengths)
of the cohomology modules in question depend polynomially on n if n « 0.
We relate our results to the examples of Singh and Katzman and discuss
some open problems.

The structure of the Rao module and the geometry of
schemes
by Marta Casanellas (Universitat de Barcelona, Spain)
Abstract:
Using a technique introduced by J. Migliore, we
will show how the structure of the Rao module of a scheme X
Ì P^{n} determines part
of the geometry of X.
In particular we will prove that if X has dimension d
³ 1
and is rBuchsbaum with r > max{codim Xd, 0},
then X is
contained in at most one variety of minimal degree.

Liaison of varieties of small dimension and deficiency modules
by Marc Chardin (CNRS & Universite Paris VI, France)
Abstract:
Liaison relates the cohomology of the ideal sheaf of a scheme to the
cohomology of the canonical module of its link. We here refer to
Gorenstein liaison in a projective space over a field: each ideal is
the residual of the other in one Gorenstein homogeneous
ideal of a polynomial ring.
Assuming that the linked schemes (or equivalently one of them) are
CohenMacaulay, Serre duality expresses the cohomology of the canonical
module in terms of the cohomology of the ideal sheaf. Therefore, in the
case of CohenMacaulay linked schemes, the cohomology of ideal sheaves
can be computed one from another: up to shifts in ordinary and homological
degrees, they are exchanged and dualized. In terms of free
resolutions this means that, up to a degree shift, they may be obtained
one from another by dualizing the corresponding complexes (for
instance, the generators of one cohomology module corresponds to the
last syzygies of another cohomology module of the link).
If the linked schemes are not CohenMacaulay, this property
fails. Nevertheless, experience on a computer shows that these
modules are closely related. We will explain in this talk this relation
for the cases of surfaces and threedimensionnal schemes.

Deformations of monomial ideals
by Aldo Conca (Università di Genova, Italy)
Abstract: There are many known ways of deforming a monomial ideal
to another ideal so that the betti numbers are preserved.
For instance, polarization, the socalled Hartshorne lifting
and algebraic shifting (of strongly stable ideals) are deformations
in this sense. I will present a simple and general deformation process
(whose special cases are polarizations and Hartshorne liftings) and discuss
various properties. This is joint work with Anna Bigatti and Lorenzo Robbiano
(Università di Genova, Italy).

Poincare series of resolution of surface singularities
by Steven Dale Cutkosky (University of Missouri, USA)
Abstract:
We associate a generally nonNoetherian graded ring to the
exceptional divisors of a resolution of singularities of
the spectrum of a two dimensional, complete normal local
ring of dimension two with algebraically closed residue
field of characteristic zero. We show that the Hilbert
polynomial of this ring is rational if the Picard group is
semiabelian, but give examples showing that it can be rational in
general.
This is joint work with Juergen Herzog and Ana Reguera.

Linearly presented ideals
by David Eisenbud (MSRI and University of California at Berkeley, USA)
Abstract:
Suppose that I is a homogeneous ideal in a polynomial ring S
in n variables over a field. Let P be the maximal homogeneous
ideal of S.
Theorem: If all the generators of I
are in degree d, and all the relations are in degree d+1,
and if I contains a power of P, then some power of I is
equal to a power of P.
Conjecture: With hypotheses as in the theorem,
I^{n1}=P^{d(n1)}.
I will explain the context and background of these ideas. Of
particular note is result relating the shape of the free
resolution of any homogeneous ideal to the monomials in the
initial ideal with respect to a reverse lexicographic ordering.
This is a preliminary report on joint work with Craig Huneke
(University of Kansas, USA) and Bernd Ulrich (Purdue University, USA).

Secant Varieties to Grassmann Varieties
By Anthony V. Geramita (Queen's University at Kingston, Canada &
Università di Genova, Italy)
Abstract:
If X Ì
P^{n} is a projective variety then by
X^{t} we will denote the closure of the set of all secant
P^{t1}'s to X, i.e.
the closure of the set of all points on linear spaces spanned by t
linearly independent points in P^{n}.
The expected dimension of
X^{t} is min{n, t dim X+(t1)}
and when this dimension is not reached the variety is defective for
(t1)secants. It is a classical problem of projective geometry to
identify the defective varieties.
We consider this problem in the case when X is a Grassmann variety. We
show how to connect this problem to one involved in finding the dimension
of a piece of an ideal in an exterior algebra. The ideal involved is of a
very special type: it is the intersection of squares of ideals generated by
families of linear forms in the exterior algebra.
We find this dimension in many cases and, as a consequence, can show (among
other things) that no secant line variety to any Grassmannian is defective
(except for the well known exceptions of the Grassmannian of two
dimensional subspaces of an (n+1)dimensional vector space). We also find
some new defective cases.
Our results use both combinatorial and commutative algebra methods.
This is joint work with: M.V. Catalisano (Genova) and A. Gimigliano (Bologna).

Positive combinatorial formulae for quiver polynomials
by Ezra Miller (MSRI and University of Minnesota, USA)
Abstract:
This talk concerns homological and geometric properties
of ideals generated by minors in products of two or more matrices
filled with independent variables. These ideals arise naturally in
representation theory of quivers, and their cohomological
invariants are polynomials that give universal topological data on
degeneracy loci for sequences of vector bundle morphisms. I will
present explicit combinatorial formulae for these cohomological
invariants. The formulae generalize in many directions the
GiambelliThomPorteous formula, which amounts to the classical
theorem that multiplicities of ideals generated by fixed size
minors in rectangles count semistandard Young tableaux. This is
joint work with Allen Knutson and Mark Shimozono.

Monomial ideals and their core
by Claudia Polini (University of Notre Dame, USA)
Abstract:
This is joint work with Bernd Ulrich (Purdue University)
and Marie Vitulli (University of Oregon).
The core of an ideal I is the intersection of all
(minimal) reductions of I. The core of a monomial ideal I
is always monomial even though I may not have any minimal reduction
which is monomial. In this talk, I will present some cases where the core of
monomial ideals has a nice geometric interpretation and it is
connected with the multiplier ideal.
I will also investigate the relationship between the core of an
ideal I and the core of the integral closure of I,
always focusing on the case of monomial ideals.

The equality I^{2}=QI in Buchsbaum rings
by Hideto Sakurai (Meiji University, Japan)
Abstract:
Let Q be a parameter ideal in a Noetherian local
ring A with the maximal ideal m and let I = Q
: m. In my talk the problem of when the equality I^{2}=QI
holds true is explored. When A is a CohenMacaulay ring, this was
completely solved by A. Corso, C. Huneke, C. Polini, and W. Vasconcelos, while
almost nothing is known when A is not a CohenMacaulay ring.
My purpose is to show that within a huge class of Buchsbaum local rings
A the equality I^{2}=QI holds true for all
parameter ideals Q and that when A is a Buchsbaum local
ring, the equality I^{2}=QI
holds true, if e(A) = 2 and depth A > 0. These results give ample
examples of ideals I, for which the Rees algebras R(I) =
Å _{n ³ 0} I^{n},
the associated graded
rings G(I) = R (I)/I R(I), and the fiber cones
F(I)= R(I)/m R(I) are all Buchsbaum rings with
certain specific graded local cohomology modules.

On birational Macaulayfications and CohenMacaulay canonical
modules
by Peter Schenzel (MartinLutherUniversität HalleWittenberg, Germany)
Abstract:
Let (A, m) denote a local domain. One motivation of
the present talk is the following question:
Does there exists a birational extension ring A Í B Í
Q, (Q denotes the field of quotients) such that B
is finitely generated as an Amodule and a CohenMacaulay ring?
We call such an extension ring a birational Macaulayfication of
A.
Theorem: Suppose that A is the factor
ring of a Gorenstein ring. Then A possesses a birational
Macaulayfication B if and only if the canonical module K(A) is
a CohenMacaulay module. Moreover, in this case B is uniquely
determined up to isomorphisms and B @
Hom_{A}(K(A),
K(A)).
This has to do with the notion of sequentially CohenMacaulay
modules (as introduced by R. P. Stanley) resp. CohenMacaulay
filtered modules (as introduced by the author). That means, the
dimension filtration {M_{i}} of a finitely generated Amodule
M (consisting of all maximal submodules M_{i} such that dim
M_{i} £
i) has the property that the quotients
M_{i}/M_{i1} are
either zero or an idimensional CohenMacaulay module. Let
K^{i}(M), 0 £
i < dim M, denote the modules of deficiency of
M, measuring the CohenMacaulay deviation of M.
Typical examples that fulfill the requirements of Theorem are affine
semigroup rings of codimension two studied by M. Morales resp. I. Peeva
and B. Sturmfels.

Ideals of linear type in Cremona maps
by Aron Simis (Universidade Federal de Pernambuco, Brasil)
Abstract:
Hulek and Schreier introduced the idea of looking into syzygies in order to
understand Cremona maps. This has now developed into a more precise theory
where other algebraic features come to play a role. I will talk about some
of these aspects and relate them to a curious question as to whether or when
the defining ideal of the Rees algebra is a minimal prime of a certain
graded algebra (besides the symmetric algebra). In the opportunity I will
revise criteria for Cremona maps and pose some further questions.

A formula for the core of ideals
by Bernd Ulrich (Purdue University, USA)
Abstract:
We give a formula for the core of certain ideals that are not necessarily
equimultiple. The core of an ideal is the intersection of all
reductions of the ideal. The concept is related to BrianconSkoda
type theorems and to a conjecture by Kawamata on the existence of
nontrivial sections of certain line bundles, as discovered recently
by Hyry and Smith.
This is joint work with Claudia Polini (University of Notre Dame).

Divisors of Integrally Closed Modules
by Wolmer V. Vasconcelos (Rutgers University, USA)
Abstract:
There is a beautiful theory of integral closure of ideals in
regular local rings of dimension two, due to Zariski, several aspects
of which were later extended to modules. Our goal is to study integral
closure of modules over normal domains by attaching
divisors/determinantal ideals to them.
They will be of two kinds: the ordinary Fitting ideal and its divisor,
and another `determinantal' ideal obtained through Noether normalization.
One or the other are useful in studying completeness, or even normality, in
some classes of modules over singular rings.
This is joint work with Jooyoun Hong (Rutgers University, USA)
and Sunsook Noh (Ewha Womans University, Korea).

Resolution of Singularities: Computational Aspects
by Orlando Villamayor (Universidad Autonoma, Spain)
Abstract:
There are two main theorem in resolution of singularities: embedded
desingularization of reduced schemes, and Logresolution of ideals
in a smooth scheme. Both theorem proved by Hironaka for the case of
excellent schemes over a field of characteristic zero.
We intend to sketch an alternative and constructive proof, always over
fields of characteristic zero, but defined in terms of an algorithm. We
will also discuss some computational aspects of the BodnarSchicho
implementation of this algorithm.

Monomial ideals and normality
by Rafael H. Villarreal (CinvestavIPN, Mexico)
Abstract: In this talk we present various recent results about
the normality of ideals and algebras associated to
monomials. Those results include obstructions for
the normality of a monomial ideal and for the normality
of a monomial subring.