2nd Bluegrass Algebra Conference
Department of Mathematics
University of Kentucky
Lexington
March 6-8, 2009
Abstracts of Talks
Geometric modeling and the Rees algebra
David Cox, Amherst College
Abstract: This talk will discuss the interesting relation between geometric modeling and the Rees algebra. I will explain how the geometric modeling community discovered the defining relations of the Rees algebra and give some examples of how these are used in curve and surface implicitization. The talk will end with a survey of some recent work on the structure of the defining relations.
Ascending chain condition for log-canonical thresholds
Lawrence Ein, University of Illinois at Chicago
Abstract: We'll discuss recent joint work with Mircea Mustata on Shokurov's ACC conjecture for log-canonical thresholds on smooth varieties.
The canonical model of a singular curve
Steven L Kleiman, Massachusetts Institute of Technology
Abstract: Given a compact Riemann surface C of genus g that is nonhyperelliptic (not a double cover of the sphere), its canonical model is an isomorphic algebraic curve C' in Pg-1 whose hyperplane sections are the divisors of zeros of its holomorphic differentials. In 1952, Rosenlicht generalized the theory to an arbitrary complete integral curve C in any characteristic.
This talk will report on some current joint work of Renato Vidal Martins and the speaker, which provides refined statements and modern proofs of Rosenlicht's results plus a determination of just when C' is rational normal, arithmetically normal, projectively normal, and linearly normal.
Defining equations of Rees algebras
Andrew Kustin, University of South Carolina
Abstract: Let I=(f1,...,fm) be an ideal in a ring R. The Rees algebra R[It] is the algebraic representation of the blow-up of Spec R along the subvariety V(I). We are interested in describing the equations which define this blow-up; in other words, the kernel of the map from R[T1,...,Tm] to R[It]. The more special I is, the more complicated this kernel becomes. We consider ideals of linear type, the Jacobian Dual, and almost linearly presented ideals of height two in k[x1,x2]. The last topic was studied with Claudia Polini and Bernd Ulrich and our interest in it is motivated by questions asked by David Cox.
Consecutive cancellations in Betti numbers of local rings
Maria Evelina Rossi, Università di Genova (Italy)
Abstract: Let I be a homogeneous ideal in a polynomial ring P over a field. By Macaulay's Theorem, there exists a lexicographic ideal L= Lex(I) with the same Hilbert function as I. Peeva proved that the Betti numbers of P/I can be obtained from the graded Betti numbers of P/L by a suitable sequence of consecutive cancellations. We extend this result to any ideal I in a regular local ring (R,n). By taking advantage of Eliahou-Kervaire's construction, from the minimal free resolution of L we can deduce results on the numerical invariants of local rings. This connection between the graded and the local knowledge is a fresh viewpoint we hope will be useful in studying the Hilbert functionsof classes of local rings.
This is a joint work with L. Sharifan, Faculty of Mathematics and Computer Science, Tehran, Iran.
Toric ideals in algebraic statistics
Sonja Petrovic, University of Illinois at Chicago
Abstract: Algebraic statistics has advanced significantly in recent years. The field focuses on the applications of algebraic geometry and its computational tools to the study of statistical models. Many interesting families of models correspond to classical families of algebraic varieties. In particular, toric ideals play a special role in the field. One family of "toric" models is given by the p1 random graph model, which models interactions in a social network. The Markov basis for this model can be understood via the special fiber ring of bipartite graphs.
Part of this research is joint work with Stephen Fienberg and Alessandro Rinaldo, of Carnegie Mellon University.
Fontaine Rings and Local Cohomology Paul Roberts, University of Utah
Abstract: Fontaine Rings were introduced in Arithmetic Geometry to give a connection between problems in mixed characteristic and corresponding problems in positive characteristic. In this talk I will define Fontaine rings, discuss their main properties, and describe some recent results on their application to some open problems in Commutative Algebra.
Geometry and syzygies of rational surfaces associated to line configurations in P2: A tale of two algebras
Hal Schenck, University of Illinois at Urbana-Champaign
Abstract: The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement, and is the quotient of an exterior algebra by a combinatorially determined ideal. The Orlik-Terao algebra is a commutative analog of the Orlik-Solomon algebra, but captures more delicate information; in particular it is not combinatorially determined. I'll discuss 2 recent results on the Orlik-Terao algebra, along with many open questions. The first result (with S. Tohaneanu) involves analyzing the tangent space to the (quadratic) Orlik-Terao algebra. The second result relates the Orlik-Terao algebra to a blowup X of P2 and certain nef but non-ample linear systems D on X, as well as syzygies of X \subset P(H0(D)).
Multiplicities and integral dependence of modules
Javid Validashti, University of Kansas
Abstract: In Equisingularity Theory, one would like to use numerical invariants to distinguish between members of a given a family of singularities. The corresponding algebraic problem leads to the study of multiplicity based criteria for integral dependence of modules, known as `Rees criteria', and generalizations of the classical notions of multiplicity. In this talk, we review some of the past results and we discuss a new approach, by introducing a multiplicity defined as a limit superior of a sequence of normalized lengths, which is a non-negative real number that can be irrational.