Midwest Commutative Algebra and Geometry Conference

View of Purdue

Purdue University
West Lafayette, Indiana
May 22 - 26, 2011

Program and Schedule

All talks will be in Math 175 (1st floor of the Math Building). Rooms Math 211 and 215 have been reserved for additional meetings/informal math discussions.
Registration will be in the Math Library Lounge (3rd floor) - also all breaks and the poster sessions will be in the Math Library Lounge.

May 22
May 23
May 24
May 25
May 26
  09:00-09:50 Ein
09:00-09:50 Migliore
10:10-10:40 Tucker
10:10-11:00 Caviglia
11:10-12:00 Mustaţă
11:30-12:20 Wiegand
Lunch Break
Lunch Break
Lunch Break
01:00-01:30 Registration
01:30-02:20 Lipman
02:00-02:50 Goto
02:40-03:30 Hartshorne
03:10-04:00 Peeva
    04:10-05:00 Poster Session 1 Poster Session 2 Poster Session 3
04:00-04:50 Rossi
05:10-06:00 Singh
05:10-06:00 Chardin

Abstracts of Talks
  1. Homological invariants of modules over contracting endomorphisms
    Luchezar Avramov, University of Nebraska
    Abstract: If φ: RR is a contracting endomorphism of a local ring R, then, viewed through φ, every non-zero finite R-module M displays the homological behavior of the residue field of R. The special case when R has positive characteristic, φ is the Frobenius endomorphism, and M=R contains a broad generalization of Kunz's criterion for regularity. The talk is based on joint work with Mel Hochster, Srikanth Iyengar, and Yongwei Yao.

  2. The Lex-Plus-Power inequality for local cohomology modules
    Giulio Caviglia, Purdue University
    Abstract: This is joint work with E. Sbarra.

  3. Cohomology and Betti numbers of graded modules
    Marc Chardin, CNRS/Université Pierre et Marie Curie, France
    Abstract: In the first part of this lecture, I will present joint results with Nicolas Botbol related to the extension to general grading of the notion of Castelnuovo-Mumford regularity. These results are very much connected to previous ones on this matter obtained by several authors. In a second part, I will present results with Tai Ha and Amir Bagheri on Betti numbers and local cohomology of powers of graded ideals.

  4. Generating sequences and semigroups of valuations
    Dale Cutkosky, University of Missouri
    Abstract: We discuss the general question of when a semigroup is the semigroup of a valuation dominating a noetherian local domain, giving some surprising examples and a necessary condition. We give a necessary and sufficient condition for the pair of a semigroup S and a field extension L/k to be the semigroup and residue field of a valuation dominating a regular local ring R of dimension two with residue field k, generalizing the theorem of Spivakovsky for the case when there is no residue field extension. We show how these methods can be used to detect the defect type of a field extension; the defect occurring in a finite projection is an essential obstruction to extending resolution of singularities to positive characteristic.

  5. Intersection multiplicity on blow-ups
    Sankar Dutta, University of Illinois at Urbana-Champaign
    Abstract: We propose a conjecture on vanishing and non-negativity of intersection multiplicity on the blow-up of a regular local ring at its closed point. Proofs of vanishing, several special cases of non-negativity and a necessary and sufficient condition for non-negativity will be presented in this talk. The connection of this conjecture with Serre's conjecture on intersection multiplicity and Peskin & Szpiro's work will also be discussed.

  6. Asymptotic syzygies of algebraic varieties
    Lawrence Ein, University of Illinois at Chicago
    Abstract: This is joint work with Rob Lazarsfeld. We discuss the shape of the minimal resolution of a smooth projective variety, when it is embedded by the complete linear system of sufficiently positive line bundles.

  7. Almost Gorenstein rings
    Shiro Goto, Meiji University, Japan
    Abstract: author's pdf file.

  8. Stable Ulrich bundles
    Robin Hartshorne, University of California at Berkeley
    Abstract: (joint work with Marta Casanellas and with the assistance of F.-O. Schreyer and F. Geiss) Many years ago Bernd Ulrich observed that the number of generators of a Cohen-Macaulay module over a local ring is bounded by the product of the rank of the module and the multiplicity of the ring. He initiated the study of modules for which this maximum occurs, now called Ulrich modules. The corresponding notion in algebraic geometry is that of an Ulrich bundle on a variety in projective space, meaning that the corresponding module over the local ring of the cone over that variety is an Ulrich module. In this paper we prove the existence of stable (hence indecomposable) Ulrich bundles of all ranks on cubic surfaces in P3 and on general cubic threefolds in P4. For bundles on the cubic surface, this requires a careful analysis of the possible curves on the surface that can be the first Chern class of an Ulrich bundle; for bundles on the cubic threefold, we need certain curves of degrees 5 and 12 whose existence (up to now) is only known via computations on Macaulay2 provided by Schreyer and Geiss.

  9. Ideals generated by quadratic polynomials
    Melvin Hochster, University of Michigan
    Abstract: Stillman has conjectured that given n forms of degree d in a polynomial ring over a field, there is a bound on the projective dimension of the ideal they generate that depends on n and d but not on the number of variables. The talk will survey what is known, and then discuss joint work of Tigran Ananyan and the speaker that solves the problem when d = 2. The bound is asymptotic to 2(n2n).

  10. Back to the future: multiplicities, integral closures, and positive characteristic
    Craig Huneke, University of Kansas
    Abstract: This talk will discuss recent joint work with Olgur Celikbas, Hailong Dao, Yi Zhang, and myself, concerning bounding Hilbert-Kunz multiplicities. A somewhat surprising point in the proofs is the appearance of the integral closure of ideals, something that takes me back to Purdue nearly 30 years ago.

  11. Complete ideals in two-dimensional regular local rings
    Mee-Kyoung Kim, Sungkyunkwan University, Korea
    Abstract: author's pdf file.

  12. Linearly presented grade 3 Gorenstein ideals in k[x,y,z]
    Andrew Kustin, University of South Carolina
    Abstract: We exhibit a monomial presentation of such ideals. This is joint work with Sabine El Khoury.

  13. Duality, residues, fundamental class
    Joseph Lipman, Purdue University
    Abstract: This talk will be about a formalism for duality, à la Grothendieck, which unifies the local theory---that is, for complete local rings or, more generally, topological rings---and the global theory---that is, for formal schemes. The formalism will be illustrated by a brief discussion of how local residue maps can be pasted together into a global map, the fundamental class. The whole topic is full of intriguing relationships, most of which have been understood for some time; but no definitive exposition has yet appeared.
    As this is a conference on commutative algebra, the talk will keep mainly to the context of commutative rings.

  14. Holonomic D-modules, a characteristic-free approach
    Gennady Lyubeznik, University of Minnessota
    Abstract: Holonomic D-modules have been studied in characteristic zero for several decades now. In characterisitc p>0 there has been no comparable theory. Recently V. Bavula has given a characteristic-free definition of holonomicity and shown that in characteristic zero it coincides with the usual notion. We will discuss some developments resulting from this approach.

  15. A tour of the weak and strong Lefschetz properties
    Juan Migliore, University of Notre Dame
    Abstract: In the first part of the talk we will give an overview of open problems and known results on the Weak and Strong Lefschetz properties. We will emphasize the many different approaches and tools that have been used to study it, behavior in positive characteristic, and connections that have been made with seemingly unrelated problems. In the second part of the talk we will describe recent work, done jointly with Rosa Miró-Roig and Uwe Nagel, on ideals generated by powers of general linear forms, and their behavior with respect to the Weak Lefschetz property.

  16. Multiplier ideals, test ideals, and ordinary varieties
    Mircea Mustaţă, University of Michigan
    Abstract: Multiplier ideals are invariants of singularities defined via divisorial valuations, while test ideals are characteristic p versions defined via the Frobenius morphism. I will discuss a conjecture relating the multiplier ideals to the test ideals via reduction mod p, and the relation of this conjecture to the ordinarity of the reductions to prime characteristic of a smooth projective variety. This is based on joint work with V. Srinivas.

  17. Resolutions over complete intersections
    Irena Peeva, Cornell University
    Abstract: This talk is on the asymptotic structure of minimal free resolutions over a graded or local complete intersection.

  18. An effective approach to the study of the local complete intersections of height two
    Maria Evelina Rossi, Università di Genova, Italy
    Abstract: Motivated by some papers of S. Goto, W. Heinzer, M.K. Kim and B. Ulrich, in this talk we present a concrete approach to the study of numerical invariants attached to a local complete intersection of height two. The underlying philosophy is that all basic non-algorithmic facts concerning Groebner bases translate into valid results on standard bases in rings of power series.

  19. Test ideals via alterations
    Karl Schwede, Penn State University
    Abstract: Given a ring R of characteristic p > 0, one can associate a test ideal which reflects subtle properties of the singularities of R. Over the last 15 years, a great deal of interest in the test ideal has been with regards to its links with multiplier ideals, similar measures of singularities often defined using complex analytic techniques. In order to compute these multiplier ideals one uses a resolution of singularities in order to change coordinates. In this talk we describe how to use alterations (partial replacements for resolutions of singularities in characteristic p > 0) to give a uniform description of an ideal which coincides with the test ideal in characteristic p > 0, the multiplier ideal in characteristic zero, and suggests a number of tantalizing questions related to the direct summand conjecture in mixed characterstic. This is joint work with Manuel Blickle and Kevin Tucker.

  20. F-pure thresholds of hypersurfaces
    Anurag Singh, University of Utah
    Abstract: The F-pure threshold is a characteristic p analogue of characteristic zero log canonical thresholds. We will discuss the calculation of F-pure thresholds for supersingular Calabi-Yau hypersurfaces.

  21. F-splitting dimension equals the dimension of the F-splitting prime
    Kevin Tucker, University of Utah/Princeton University
    Abstract: When working in equal characteristic p > 0, splittings of the Frobenius endomorphism (and its iterates) have long been used to study singularities. To that end, the F-splitting dimension of a local ring gives the asymptotic growth of the number of such splittings. When this invariant was introduced by I. Aberbach and F. Enescu, they further questioned whether it was equal to the dimension of a certain naturally defined prime ideal (the F-splitting prime). In this talk, I will describe joint work in progress with M. Blickle and K. Schwede showing that this question has a positive answer. The main ingredient in the proof is the construction of generalized F-signature (following the generalizations of tight closure and test ideals to incorporate ideal and divisor pairs). Time permitting, we hope to mention some other applications and examples of this beautiful and still developing theory.

  22. Comparing powers and symbolic powers of ideals
    Javid Validashti, University of Kansas
    Abstract: Comparing the ordinary and symbolic powers of ideals and giving criteria for equality are subjects of interest in both commutative algebra and algebraic geometry. We show that the symbolic topology defined by a prime ideal is uniformly linearly equivalent to the adic topology for a large class of isolated singularities. We also consider the following question posed by Huneke on the equality. In a regular local ring, if the ordinary and symbolic powers of a prime ideal are the same up to its height, then are they the same for all powers? We provide supporting evidence for ideals defining classes of monomial curves and for prime ideals defining rings with low multiplicity. This talk is based on joint works with Aline Hosry, Youngsu Kim, Craig Huneke and Dan Katz.

  23. The homology of parameter ideals
    Wolmer Vasconcelos, Rutgers University
    Abstract: author's pdf file.

  24. Certain jet schemes have rational singularities
    Kei-ichi Watanabe, Nihon University, Japan
    Abstract: author's pdf file.

  25. Power series over Noetherian rings
    Sylvia Wiegand, University of Nebraska
    Abstract: This is joint work with William Heinzer and Christel Rotthaus. We show how to create exotic Noetherian and non-Noetherian rings using power series over well-understood Noetherian commutative integral domains. This construction synthesizes a classical technique used by Akizuki in the nineteen thirties and by Nagata in the nineteen fifties. Usually we simply intersect an appropriate field with the power series ring over the base Noetherian domain. We have been able to show that in certain circumstances, such an intersection is computable as a directed union, and the Noetherian property for the associated directed union is equivalent to a flatness condition. This flatness criterion simplifies the analysis of several classical examples and yields new examples.
    We also describe some examples produced using the technique.