Midwest Commutative Algebra and Geometry Meeting: |
A Conference in Honor of Joseph Lipman
Purdue University, West Lafayette
May 17-21, 2004
List of Talks
Some applications of positive characteristic techniques to
by Donu ARAPURA, Purdue University
I will discuss how the method of Deligne-Illusie leads to both
new results and to new proofs of old theorems. As example of the
latter, I will indicate a new proof of a theorem of Simpson about
the structure of cohomology support loci in the Picard group.
Reverse homological algebra over some local rings
by Luchezar AVRAMOV, University of Nebraska at Lincoln
Abstract. When R is a noetherian local ring, k
is its residue field, and M is an R-module,
Yoneda products turn E = ExtR(k,k)
into a graded algebra and ExtR(M,k) into a graded left
E-module. We consider the question: when is a graded left
E-module M the cohomology of some finitely
generated R-module? Satisfactory answers are available in certain cases,
including complete intersection rings, Gorenstein rings of low
codimension, and Golod rings. This is joint work with Dave Jorgensen.
Equiresolution of quasi-ordinary surface singularities
by Chunsheng BAN, Ohio State University
J. Lipman outlined an approach to prove that Zariski equisingularity
in dimensionality type 2 implies equiresolvability. This approach
reduces the general case to the case of quasi-ordinary singularities.
We will investigate equiresolution in the case of an equisingular
family of quasi-ordinary surface singularities.
Resolution of singularities of toric varieties
by Edward BIERSTONE, University of Toronto
I will discuss a combinatorial algorithm for equivariant
embedded resolution of singularities of a toric variety
over a perfect field. Desingularization is realized by
a finite succession of blowings-up with centres that are
orbit closures and satisfy the normal flatness criterion
of Hironaka. (Joint work with Pierre Milman.)
Grothendieck residues à la Lipman and the centre of the
by Ragnar-Olaf BUCHWEITZ, University of Toronto
For any algebra there is a canonical homomorphism of graded commutative
rings from its Hochschild cohomology to the graded centre of its derived
category. The definition of that homomorphism is conceptual but does not
say anything about its properties such as injectivity or surjectivity. We
show that Lipman's interpretation of Grothendieck residues of finite flat
representations of the algebra allows to bound the kernel of that
homomorphism. In fact, the bounds so obtained are sharp in the few
examples we can effectively calculate. In particular, the homomorphism
from Hochschild cohomology to the graded centre of the derived category
turns out to be injective for affine smooth algebras over a field.
Poincare Series, arithmetic and geometry of singularities
by Antonio CAMPILLO, Universidad de Valladolid
Given a noetherian local domain and finitely many divisorial valuations
centered at it, one has an associated natural multifiltration (given by
integer valuation conditions) which gives rise to a naturally associated
Poincare series. In geometrical context, for given singularities, those
Poincare series can be understood as concrete with respect to Euler
characteristics on precise function spaces. We discuss several cases for
which the computation of such integrals is available, giving rise to some
expressions relating topological and geometrical features of the singularities
to arithmetical properties of Poincare series of their local rings.
Some peculiar behavior in positive characteristic
by Brian CONRAD, University of Michigan
It is extraordinarily difficult to prove anything about the
primality behavior for specialization of irreducible polynomials over the
integers, and it is also very hard to unconditionally prove anything about
the statistical behavior of root numbers in non-isotrivial 1-parameter
families of elliptic curves over the rationals, though there are standard
conjectures in both situations; the difficulties are caused by hard
problems concerning the Mobius function. Rather surprisingly, when one
replaces the integers with the coordinate ring of a smooth affine curve of
arbitrary genus over a finite field (with one rational point at infinity),
the Mobius function exhibits unexpected periodicity behavior;
consequently, in the study of irreducible specialization and root-number
variation one finds new phenomena unlike anything believed to occur in the
classical case. We will discuss some interesting theorems that can be
proved for the function-field problems of irreducible specialization
and/or variation of root numbers, and if time permits we will also discuss
some of the vexing difficulties that arise for irreducible specialization
in characteristic 2.
Multiplicity of special fiber rings and Sally modules
by Alberto CORSO, University of Kentucky
The special fiber ring F(I) of an R-ideal I encodes various asymptotic
information about the ideal I. In addition, Proj(F(I)) corresponds to the
fiber over the closed point of the blowup of the variety Spec(R) along
the subvariety V(I).
In this talk we will describe sharp bounds on the multiplicity of
F(I) involving the Hilbert coefficients of I. We will show that
equality in these bounds reflect on `good' structural properties of F(I).
These results, in a Cohen-Macaulay or Buchsbaum setting, are based on a
careful analysis of the properties of a related graded object:
the Sally module.
Toroidalization of projective morphisms of 3-folds
by Steven Dale CUTKOSKY, University of Missouri
We discuss our proof that birational morphisms of projective 3-folds
can be made toroidal by blowing up nonsingular subvarieties in the domain and
Some aspects of positivity and non-negativity of intersection
by Sankar DUTTA, University of Illinois, Urbana
In this talk we would like to present the following:
(1) a special case of positivity of Serre's conjecture on intersection
multiplicity; (2) some aspects of non-negativity of intersection multiplicity
on complete intersections.
Small algebraic sets
by David EISENBUD, MSRI and University of California, Berkeley
Famous theorems of Del Pezzo and Bertini classify the
nondegenerate varieties X in a projective space over an algebraic
closed field such that deg X has the minimal possible value,
deg X = 1 + codim X. These are also the varieties of
Castelnuovo-Mumford regularity 2. It turns out that the classification can
be extended to all reduced schemes of regularity 2, and these have a geometric
characterization similar to `minimal degree.' I will report on these
results, which are the subject of joint work with Mark Green, Klaus
Hulek, and Sorin Popescu.
Connectedness theorems for degeneracy loci over local rings
by Hubert FLENNER, University of Bochum
This is a report on joint work with B. Ulrich. Let (A, m) be a local
noetherian ring and M, N modules of rank m,
n, respectively, which have an isolated singularity, i.e. M and N localized
at p are free for all primes
p not equal to m. For an A-linear map f from M to
N we consider its r-th degeneracy locus Dr(f) which is the set of
all primes p belonging to Spec(A) with rk(f(p)) at most r,
where f(p) denotes the map f tensored with the residue field of p.
In this talk we discuss connectedness theorems
for Dr(f). For instance we show: If A is a complete integral
domain and f belongs to m Hom(M,N) then Dr(f) is
connected in dimension dim A-(n-r)(m-r).
In the particular case M=N=A we recover Grothendieck's connectedness
theorem. Similar results hold for symmetric and alternating maps of modules.
We also discuss the relations with connectedness results of Fulton-Lazarsfeld
and F. Steffen.
Betti numbers of semi- and subalgebraic sets
by Andrei GABRIELOV, Purdue University
A review of recent results on the upper bounds
for the Betti numbers of real semialgebraic sets,
projections of semialgebraic sets, and Hausdorff
limits of semialgebraic families.
Gorenstein liaison of projective varieties
by Robin HARTSHORNE, University of California, Berkeley
While liaison (linkage) has been used in algebraic geometry and
commutative algebra for some time, defined using complete intersection
subschemes, the newer notion of Gorenstein liaison shows considerable
promise and raises interesting questions. In this talk I will discuss
some open problems and new results in this rapidly developing field.
Singularities: a promenade
by Herwig HAUSER, Universität Innsbruck
Our sightseeing tour through singularity land will lead us to
several creatures exhibiting different geometric features:
transversality, triviality, symmetry, deformation, blowup, projection.
Though it is forbidden to touch or feed the animals, we will at least
try to have a close view on them.
Thirteen questions in Commutative Algebra
by Melvin HOCHSTER, University of Michigan
The talk will discuss the status of a substantial number of significant
open problems. Some have been intractable for decades, while others are
more recent or even virtually new. The conjectures selected come from a
wide range of subareas of the field.
Homomorphisms for injective hulls
by I-Chiau HUANG, Academia Sinica
Injective hulls are building blocks of injective resolutions. If we have models
for them, descriptions of the homomorphisms for these models in the resolutions
give rise to cohomological information. Lipman's idea of pseudofunctors on
modules with zero dimensional support provides a relatively canonical way
to construct injective hulls, hence a framework for working out the coboundary
maps of injective resolutions. In the talk, we illustrate these ideas
examples including semigroup rings, Akizuki's one dimensional local domain,
homogeneous coordinate ring of a special quadric surface of the projective
space, and vector bundles on projective spaces.
Equisingular Strata of Curves and Clusters on Surfaces
by Steven KLEIMAN, MIT
Consider a family of reduced curves on a smooth complex surface, and
stratify the parameter space by the singularity type of the curves.
These strata are known to be locally closed. But, in fact, their
components carry natural multiplicities of importance in enumerative
questions. These multiplicities appear through a study of related
subschemes H(D) of the Hilbert scheme. Each H(D)
parameterizes the clusters defined by the complete ideals of a given
singularity type D. Each H(D) is, a priori, nonreduced,
but turns out to be smooth. This is joint work with Ragni Piene.
A surprising fact about D-modules in characteristic p>0
by Gennady LYUBEZNIK, University of Minnesota
Let k be a field of characteristic p>0, let R be the ring
of polynomials in a finite number of variables over k, let D be the
ring of k-linear differential operators of R and let f be a non-zero
polynomial in R. We prove that R[1/f] obtained from R by inverting f
is generated as a D-module by 1/f. This is a striking fact
considering that counterexamples to an analogous statement in
characteristic 0 have long been known. This is joint work with Josep
Alvarez Montaner of the University of Barcelona, Spain.
by Juan MIGLIORE, University of Notre Dame
I will discuss joint work with Uwe Nagel. Consider a space curve
whose definining ideal is an intersection of
powers of monomial prime ideals of height two. There are six such prime
ideals, and the curve is equidimensional, and supported on a
tetrahedral configuration of lines. We are interested in the even
liaison class of such a curve, and in particular, when it is
arithmetically Cohen-Macaulay (ACM). Schwartau described when certain
such curves are ACM, namely he restricted to curves supported on a
certain four of the six lines.
We first show that starting with an arbitrary tetrahedral curve, there is
a particular reduction that produces a smaller tetrahedral curve and
preserves the even liaison class. We call the curves that are minimal
with respect to this reduction S-minimal curves. Given a tetrahedral
curve, we describe a simple algorithm (involving only integers) that
computes the S-minimal curve of the corresponding even liaison class; in
the process it determines if the original curve is arithmetically
Cohen-Macaulay or not. We also describe the minimal free resolution of
an S-minimal curve, using the theory of cellular resolutions. This
resolution is always linear. This result allows us to classify the
arithmetically Buchsbaum, non-ACM tetrahedral curves. More importantly,
it allows us to conclude that an S-minimal curve is minimal in its even
liaison class, in the sense of the Lazarsfeld-Rao property. Finally, we
show that there is a large set of S-minimal curves such that each curve
corresponds to a smooth point of a component of the Hilbert scheme and
that this component has the expected dimension.
Minimal resolution of relatively compressed level algebras
by Uwe NAGEL, University of Kentucky
A graded Artinian algebra of socle degree vector s and
codimension c is said
to be compressed if its Hilbert function is the maximal possible,
given s and c. Generalizing this, if I is a complete
intersection ideal in the polynomial ring R in n variables
and the s is given, then an Artinian quotient A of
R/I is relatively compressed (with respect to I)
with socle degrees s if the Hilbert function of A is
maximal among all Artinian quotients with the same socle degrees.
We study the Hilbert function and the minimal resolution of such
algebras when I is a general complete intersection. In particular,
we relate these problems to Froeberg's
conjecture about the Hilbert function of ideals generated by general
forms and to the Minimal Resolution Conjecture of Mustata
for points on a projective variety. Moreover, we find
situations where ``ghost'' terms in the resolution are forced to
exist, thus the graded Betti numbers are not those one would expect
from the Hilbert function. This is not surprising when these ghost
terms result from Koszul relations, but we show that they can come for
other reasons, too.
Finally, we use our results to find the minimal free resolution of
general Artinian almost complete intersections in many new cases.
This is joint work with J. Migliore and R. Miro-Roig.
Integral closure filtrations
by Claudia POLINI, University of Notre Dame
This is joint work with Bernd Ulrich, Wolmer Vasconcelos, and Rafael
Villarreal. Using a generalized version of the Briançon Skoda theorem,
we give a tight upper bound for the first Hilbert coefficient of the
integral closure filtration of an ideal. This bound is linear in the
multiplicity. It provides a measure for the complexity of the process
used to compute the integral closure of the Rees algebra.
Local cohomology and the homological conjectures
by Paul ROBERTS, University of Utah
One of the standard techniques in the study of the homological conjectures
has been to exploit vanishing properties of local cohomology. A couple of
years ago Heitmann proved an `almost vanishing' theorem for local cohomology
for rings of mixed characteristic in dimension three and used it to prove
the Direct Summand Conjecture in that case. In this talk I will
discuss the connection between this type of almost vanishing
property and the homological conjectures, present some related
questions that are interesting also for rings of characteristic
zero, and describe some recent joint results
with V. Srinivas where these questions can be answered.
From Serre duality on varieties to duality on formal schemes
by Pramathanath SASTRY, University of Toronto
The talk will be an overview of topics in duality beginning with
Joe Lipman's blue book (Asterisque 117),
touching on themes related to Lipman's research on duality and
ending with recent work on duality on formal schemes by various
people mathematically close to Lipman.
Tracking numbers of graded algebras
by Wolmer VASCONCELOS, Rutgers University
We introduce the technique of tracking numbers of graded algebras and
modules. It is a modified version of the first Chern class of its free
resolution relative to any of its standard Noether normalizations.
Several estimations are obtained which are used to bound the length
of chains of algebras occurring in any construction of the integral
closure of a graded domain. Noteworthy is a quadratic bound on the
multiplicity for all chains of algebras that satisfy the condition
S2 of Serre. (This is joint work with Kia Dalili.)
Resolution, equi-resolution, and stratification of families
by Orlando VILLAMAYOR, Universidad Autonoma de Madrid
We will discuss first the Theorems of Embedded Principalization
of Ideals, and of Embedded Resolution of Schemes, both over fields of
characteristic zero. We will then introduce the notion of equisolvable
families of scheme, and give some indications on how to stratify arbitrary
families of schemes into equisolvable families.
In fact, the algorithmic proofs of the first two theorems allow us to
classify schemes according to their resolution of singularities. This has
applications, for example, to the universal families obtained with the aid
of the Hilbert Scheme Theory, which can be stratified into universal
The behavior, in this context, of some invariants arising from Motivic
Integration Theory will also be mentioned.
Writing equations of surface singularities from their topology
by Jonathan WAHL, University of North Carolina
Via resolution, any complex normal surface singularity X gives rise to a
negative-definite configuration E of exceptional curves, and E is essentially
equivalent to the topology of X. Any E can occur. But, given E, can one write
down equations for some X? In joint work with Walter Neumann, we study the
case that E is a tree of rational curves ("the link is a rational homology
sphere"). This case is MUCH more general than having a "rational singularity",
including hypersurface singularities of arbitrarily large degree. Using an
auxiliary "splice diagram," we show how in many cases to write down such an X
explicitly as the quotient a complete intersection singularity by a finite
abelian group. That is, we find the "universal abelian covering". Understanding
exceptional cases would require understanding an analogue of the "Hilbert class
field" of algebraic number theory.
Cohomological properties of families of hypergeometric systems
by Uli WALTHER, Purdue University
Let T be an algebraic torus acting diagonally on
V = Cn.
Gelfand, Graev, Kapranov and Zelevinsky introduced in the 80's
a holonomic system of partial differential equations from this torus
action depending on a complex parameter b,
the so-called A-hypergeometric system
It consists of Euler operators expressing homogeneity
conditions, and binomial operators with constant coefficients reflecting
the underlying toric variety.
It turns out that the rank, the number of holomorphic solutions,
of this system equals for most b
the volume of the convex hull of the
columns of the matrix A inducing the T-action, and the origin.
Since the introduction of HA(b)
many authors have worked
on predicting when rank and volume agree: Adolphson, Sturmfels, Takayama,
Cattani, Dickenstein, D'Andrea, Saito, Matusevich, Miller.
I explain the precise structure of the set of b
for which the volume does not give the number of solutions. They are
determined by the local cohomology groups of the toric variety associated
to the torus action induced by A. The connection between these two
objects is given by a Koszul-like complex induced by the Euler operators
which relates rank jumps to the failure of flatness of the system at
a parameter, and local duality.
In particular, absence of rank jumps is equivalent to Cohen-Macaulayness
of the toric variety, and the set of rank jumping parameters is the
finite union of shifted Zariski closures faces of the semigroup
This is joint work with Laura Matusevich and Ezra Miller.
Some results on multiplier ideals and integrally closed
by Keiichi WATANABE, Nihon University
We will discuss two problems concerning multiplier ideals.
(1) Subadditivity; the "subadditivity" property for multiplier
ideals in regular rings is a very important property. But the proof
depends heavily on regularity and cannot be generalized to the
non-regular case. We show that the subadditivity property holds in log
terminal singularities in dimension 2. We also give counterexamples in
dimension 3, which show that "log terminal" is not sufficient for
subadditivity to hold. This is joint work with Shunsuke Takagi.
(2) Given an integrally closed ideal I, we ask if there exists
some ideal L and a rational number s such that I = J(Ls), where J
denotes the multiplier ideal. We give an affirmative answer in case the
ring is a 2 dimensional log terminal singularity with algebraically
closed residue field. This is joint work with J. Lipman.
Simple Hironaka resolutions
by Jaroslaw WLODARCZYK, Purdue University
Building upon work of Villamayor and Bierstone-Milman we give a proof
of the canonical Hironaka principalization and desingularization.
We introduce here the idea of homogenized ideals which gives a
priori the canonicity of algorithm and radically simplifies the proof.