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 vanishing theorems
    by Donu ARAPURA, Purdue University
    Abstract: 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: 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
    Abstract: 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
    Abstract: 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 derived category
    by Ragnar-Olaf BUCHWEITZ, University of Toronto
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: We discuss our proof that birational morphisms of projective 3-folds can be made toroidal by blowing up nonsingular subvarieties in the domain and target.

  • Some aspects of positivity and non-negativity of intersection multiplicity
    by Sankar DUTTA, University of Illinois, Urbana
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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 by concrete 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
    Abstract: 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
    Abstract: 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.

  • Tetrahedral curves
    by Juan MIGLIORE, University of Notre Dame
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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
    Abstract: 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 of schemes
    by Orlando VILLAMAYOR, Universidad Autonoma de Madrid
    Abstract: 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 equisolvable families. 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
    Abstract: 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
    Abstract: 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 HA(b). 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 NA. This is joint work with Laura Matusevich and Ezra Miller.

  • Some results on multiplier ideals and integrally closed ideals
    by Keiichi WATANABE, Nihon University
    Abstract: 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
    Abstract: 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.