3^rd Bluegrass Algebra Conference
 
Department of Mathematics
University of Kentucky
Lexington
June 14-16, 2012



Program and Schedule

All talks will be held in room 208 of the Classroom Building (CB 208)
(next to the Patterson Office Tower)

Refreshments will be served in room 745 of the Patterson Office Tower (POT 745)

Thursday June 14 Friday June 15 Saturday June 16       09:30-10:20   Hochster   Cutkosky         10:20-11:10 Coffee and Tea   Coffee and Tea         11:10-12:00 Nagel   Niu           Lunch Break             Lunch Break       02:00-02:50 Goto   Heinzer         03:00-04:00 Coffee and Tea   Coffee and Tea   Coffee and Tea           04:00-04:50 Ein   Lin   Xie           05:00-05:50 Ha   Walther   Schenck           07:30-09:30 Banquet (Masala Indian Restaurant)

Abstracts

1. Asymptotic Multiplicities
   Dale Cutkosky, University of Missouri
   Abstract: Suppose that (R, m) is a local ring of dimension d, and I_i for i ∈ Z₊ is a family of ideals in R such that I_iI_j ⊂ I_i+j for all i,j. We discuss asymptotic multiplicities associated to this family. A consequence of our main result is the following theorem:
   Theorem 1. Suppose that R is a regular local ring or a normal, excellent local ring of characteristic zero. Let s=limsup_{n → ∞} {dim(R/I_n)}. Then lim_{n → ∞} e_s(m,R/I_n)/n^d-s ∈ R exists. Here e_s(m,R/I_n) is the modified multiplicity of Serre.
   As a consequence of our main result, we obtain the following theorem which gives a positive answer to a question raised in a recent paper by the author, Kia Dalili and Olga Kashcheyeva:
   Theorem 2. Suppose that R is a regular local ring or an excellent normal local ring (of any characteristic). Further suppose that R contains an algebraically closed field which is isomorphic to its residue field. Let ν be a rank one valuation of the quotient field of R which dominates R. Let S^R(ν) be the semigroup of values of elements of R, which can be regarded as a sub semigroup of R₊. For n ∈ Z⁺, define φ(n)=|S^R(ν)∩ (0,n)|. Then lim_{n → ∞} φ(n)/n^d exists.
   We discuss the relationship of our results to recent work by Okounkov, Mustata, Lazarsfeld, Fulger and others.
2. Jacobian multiplier ideals
   Lawrence Ein, University of Illinois at Chicago
   Abstract: This is joint work with Ishii and Mustata. We define a version of multiplier ideals using Jacobian discrepancies. We prove a version of the relative vanishing theorem and obtain a versoin of Briancon-Skoda type formula for singular varieties.
3. Almost Gorenstein rings: an attempt toward higher dimensional cases
   Shiro Goto, Meiji University, Japan
   Abstract: In the previous paper [GMP] the author, N. Matsuoka, and T. T. Phuong gave an alternative definition of one-dimensional almost Gorenstein local rings and developed a theory about them. Almost Gorenstein rings were originally introduced by V. Barucci and R. Fröberg [BF] in the case where the local rings are of dimension one and analytically unramified. They developed in [BF] a very nice theory of almost Gorenstein rings and gave many interesting results as well. The paper [GMP] aimed at an alternative definition of almost Gorenstein ring which can be applied also to the rings that are not necessarily analytically unramified. One of the purposes of such an alternation is to go beyond a gap in the proof of [BF, Proposition 25] and solve in full generality the problem of when the algebra m : m is a Gorenstein ring, where m denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one.
   The present purpose is to search for possible definitions of higher-dimensional almost Gorenstein local/graded rings. I will give a candidate in terms of embedding of the base Cohen-Macaulay rings into their canonical modules. I will confirm that several results of one-dimensional case are safely extended to those of higher dimensional cases. Examples and open problems will be discussed also.
   [BF]    V. Barucci and R. Fröberg, One-dimensional almost Gorenstein rings, J. Algebra, 188 (1997), 418-442.
   [GMP] S. Goto, N. Matsuoka, and T. T. Phuong, Almost Gorenstein rings, Preprint (2011).
4. Powers of ideals in combinatorics
   Tai Ha, Tulane University
   Abstract: We survey recent studies relating powers of ideals and combinatorial structures. In particular, we describe how to detect important invariants and properties of (hyper)graphs by solving ideal membership problems and computing associated primes. In addition, we discuss the equivalence between the Conforti-Cornu\’ejols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.
5. The Rees valuations of complete ideals in a regular local ring
   Bill Heinzer, Purdue University
   Abstract: I am reporting on joint work with Mee-Kyoung Kim of Sungkyunkwan University. In the case where I is a complete m-primary ideal of a regular local ring (R,m) of dimension two, the beautiful theory developed by Zariski about complete ideals of R implies that the Rees valuation rings V of I are in a natural one-to-one correspondence with the minimal primes P of the ideal IR[It] in the Rees algebra R[It]. In work of Huneke and Sally and then in later work of the authors, the structure of R[It]/P is considered in the case where the residue field R/m = k is relatively algebraically closed in the residue field of V. I will discuss the structure of R[It]/P without this assumption that k is relatively algebraically closed in the residue field of V.
   In a regular local ring of dimension greater than two, it is known that many of the properties of complete ideals in the dimension two case no longer hold. Thus it is natural to restrict to special classes of complete ideals such as the finitely supported complete ideals considered by Joseph Lipman or the monomial complete ideals considered by John Gately. I will discuss some properties of the Rees valuations of complete ideals in these cases.
6. Strong F-regularity, F-splitting, and small Cohen-Macaulay modules over multi-graded rings
   Mel Hochster, University of Michigan
   Abstract: The talk will describe recent work of Yongwei Yao and the speaker which gives new characterizations of strongly F-regular rings, new splitting theorems for modules over rings with a module-finite strongly F-regular extension, and also shows that rings of positive characteristic with a multi-grading have small multi-graded Cohen-Macaulay modules under certain conditions that are milder than previously known. For example, every finitely generated bi-graded (by N²) domain R of Krull dimension 4 over a field K of positive characteristic such that the R_(0,0) = K has a small, maximal bi-graded Cohen-Macaulay module. (The bi-grading must be ``honest" in the sense that there are nonzero graded pieces in degree pairs that are linearly independent over the rational numbers.) There are similar results for gradings over N^k.
7. Defining ideals of Rees algebras as divisors
   Kuei-Nuan Lin, University of California at Riverside
   Abstract: This is joint work with C. Polini. The Rees algebra of an ideal provides an algebraic realization for the classical notion of blowing up a variety along a subvariety, which is a fundamental operation in algebraic geometry and commutative algebra. Understanding the defining ideal of a Rees algebra is difficult in general. In collaboration with Polini, we have been able to compute the defining ideals of Rees algebras of direct sums of powers of the maximal ideal of a polynomial ring. In this case, the Rees algebras are normal domains. We further treat the case of ideals in a polynomial ring defined by forms of the same degree that arise as truncations of regular sequences of length two. We think of these defining ideals as divisors of normal domains. In this context, we study the Cohen-Macaulayness and regularity of Rees algebras as well.
8. Blow-up rings associated to Ferrers and threshold graphs
   Uwe Nagel, University of Kentucky
   Abstract: Starting from the classical Dedekind-Mertens lemma about the content of polynomials, we consider various rings and ideals associated to Ferrers and threshold graphs. This includes minimal reductions, special fiber and Rees rings. In particular, a generalization of ladder determinantal ideals of a symmetric matrix is studied.
   This is joint work with Alberto Corso, Sonja Petrovic, and Cornelia Yuen.
9. Singularities of generic linkages of algebraic varieties
   Wenbo Niu, Purdue University
   Abstract: Let A be either a nonsingular affine variety or a projective space over the field of complex number. Consider a closed subvarieties X of A. If we choose equations carefully among the defining equations of X then we can obtain a compete intersection V containing X and a subscheme Y such that Y is linked to X via V. If we choose such a complete intersection V as general as possible, then Y is called a generic link of X.
   The study of linkage, or the theory of liaison, of algebraic varieties can be tracked back to hundred years ago. In this talk we are trying to provide a modern approach to study the singularities of generic linkage. We shall show how resolution of singularities and multiplier ideal sheaves can be used in the generic linkage theory and how we study this theory from the viewpoint of pairs. As a consequence of our approach, we shall give some results on singularities preserved by generic linkages.
10. Syzygies and singularities of tensor product surfaces
    Hal Schenck, University of Illinois at Urbana-Champaign
    Abstract: Let U ⊆ H⁰(O_{P¹ × P¹}(2,1)) be a basepoint free four-dimensional vector space. The sections corresponding to U determine a regular map φ_U: P¹ × P¹ → P³. We study the associated bigraded ideal I_U ⊆ k[s,t;u,v] from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation for φ_U(P¹ × P¹), via work of Busé-Jouanolou, Busé-Chardin, Botbol, and Botbol-Dickenstein-Dohm on the approximation complex Z. We prove that the existence of pure first syzygies (syzygies of bidegree (a,0) or (0,a)) allows a substantial simplification in the computation of the implicit equation: in all but one case, the implicit equation is a specific minor of the first differential in Z. (Joint work with A. Seceleanu and J. Validashti)
11. Jacobian ideals and D-invariants
    Uli Walther, Purdue University
    Abstract: We identify some roots of the Bernstein-Sato polynomial using information on the Jacobian ideal of a homogeneous divisor. In the process we also find some cohomology classes on the corresponding Milnor fiber.
    Some natural conjectures in D-module theory are discussed which give rise to interesting open problems in commutative algebra.
12. The generalized multiplicities and Hilbert functions
    Yu Xie, University of Notre Dame
    Abstract: This talk focuses on the recent development of generalized multiplicities and Hilbert functions and its applications to the study of graded algebras, index of nilpotency, normalizations of ideals, etc.