Commutative Algebra Seminar

In the study of quadratic moment maps there appears to be a dichotomy between large and small representations. In the large case the moment map exhibits some features of regularity. This enables one to make qualitative and quantitative statements about the symplectic quotients. In the talk I will report on recent joint work with Gerald Schwarz and Christopher Seaton which proves that for 2-large representations the appropriately defined complex symplectic quotient has symplectic singularities. This in particular entails that the symplectic quotient is graded Gorenstein domain and is a normal variety with rational singularities. Furthermore, using a recent theorem of of P. Etingof and T. Schedler, it follows as a corollary that in the 2-large case the space of Poisson traces is finite dimensional. It is expected that the result generalizes to small representations as well, even though here the moment map can have all sorts of pathologies. For this conjecture it is crucial to use the definition of the symplectic quotient that involves the real radical of the ideal generated by the moment map.

The Gabber-Gillet-Suslin-Thomason rigidity theorem states that for a henselian pair (R, I) with p invertible on R, the mod p algebraic K-theory of R and R/I agree. We prove a generalization of this result for arbitrary henselian pairs, where the difference is measured by topological cyclic homology; when I is nilpotent this is due to Dundas-McCarthy. We recover several computations in p-adic algebraic K-theory and obtain some new structural results, e.g., on continuity in K-theory. This is joint work with Dustin Clausen and Matthew Morrow.

In 2004, Dwyer, Greenlees, and Iyengar gave a necessary condition, on each homologically finite complex, for a local ring to a be a complete intersection. The main goal of the talk is to give an outline of the proof that the converse holds. To do this, I will introduce the notion of Koszul support varieties which is the main ingredient in establishing this result.

The Frobenius endomorphism of a local ring of prime characteristic gives rise to Frobenius actions on local cohomology modules. In this talk we will discuss interesting connections between these Frobenius actions, tight closure, and Frobenius closure. All new results presented are from joint work with Pham Hung Quy.

I will discuss cohomological obstructions to rationality, descent varieties, and unirationality questions.

Given a commutative Noetherian ring R, we are interested in the cohomology annihilator ideal ca(R). This ideal consists of the ring elements which annihilate all Ext-modules Ext^n(M,N) for any finitely generated R-module M,N and sufficiently large n. It is closely related to the singularities of our commutative ring. In this talk, we will give the necessary background and show that in dimension 1, under reasonable assumptions, the cohomology annilator ideal is the conductor ideal. If time permits, we will investigate the relation between the cohomology annihilator and the stable annihilator of a noncommutative crepant resolution of R.

A vector bundle on a projective variety has natural cohomology if every twist has cohomology concentrated in a single degree. Such vector bundles form extremal rays in the cone of cohomology tables of vector bundles. This cone was studied by Eisenbud and Schreyer in connection with Boij-Soederberg theory. Eisenbud and Schreyer considered the natural generalization of the cone of cohomology tables to the bi-graded setting of P1 x P1. They conjectured that there should exist vector bundles on P1 x P1 with natural cohomology with essentially prescribed Hilbert polynomial. In this talk I'll state the conjecture precisely and prove it in "most" cases.

We present results on the geometry and cohomology of effective divisors on a smooth cubic surface in P3. We study the linear systems using Zariski decomposition, and determine their cohomologies. Furthermore, we study the Hartshorne-Rao modules of the curves, and determine the degrees of their generators. Altogether, we show how to determine the free resolution of such curves from counting of secant lines. The results are based on the work of Maggioni and Giuffrida, but include corrections to several main theorems as well as simplification and generalization.

The classical volume polynomial in algebraic geometry measures the degrees of ample (and nef) divisors on a smooth projective variety. We introduce an analogous volume polynomial for matroids, and give a complete combinatorial formula. For a realizable matroid, we thus obtain an explicit formula for the classical volume polynomial of the associated wonderful compactification. We then introduce a new invariant called the volume of a matroid as a particular specialization of its volume polynomial, and discuss its algebro-geometric and combinatorial properties in connection to graded linear series on blow-ups of projective spaces.

For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds on a fixed strongly F-regular germ satisfies the ascending chain condition. This is a positive characteristic analogue of the "ascending chain condition for log canonical thresholds" in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.