Commutative Algebra Seminar
Spring 2018, Friday 2:30–3:20, LCB 222
Date  Speaker  Title — click for abstract 
January 12  Florian Enescu Georgia State University 
Frobenius complexity in the graded setting

January 19  HansChristian Herbig CEP, Brazil 
Moment map phenomenology
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 2large 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 2large 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.

January 26  Steven V Sam University of Wisconsin, Madison 
Special Colloquium

February 16  Akhil Mathew University of Chicago 
Rigidity in algebraic Ktheory and topological cyclic homology
The GabberGilletSuslinThomason rigidity theorem states that for a henselian pair (R, I) with p invertible on R, the mod p algebraic Ktheory 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 DundasMcCarthy. We recover several computations in padic algebraic Ktheory and obtain some new structural results, e.g., on continuity in Ktheory. This is joint work with Dustin Clausen and Matthew Morrow.

February 23  Josh Pollitz University of Nebraska, Lincoln 
A characterization of complete intersections in terms of the derived category
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.

March 2  Thomas Polstra University of Utah 
On the nilpotence of Frobenius actions on local cohomology modules
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.

March 9  Yuri Tschinkel NYU, Courant 
Rationality and Unirationality
I will discuss cohomological obstructions to rationality, descent varieties, and unirationality questions.

April 6  Ozgur Esentepe University of Toronto 
Cohomology annihilators in dimension one
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 Extmodules Ext^n(M,N) for any finitely generated Rmodule 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.

April 6  Pablo Solis Stanford University 
Hunting Vector Bundles on P1 x P1
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 BoijSoederberg theory. Eisenbud and Schreyer considered the natural generalization of the cone of cohomology tables to the bigraded 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.

April 13  Mengyuan Zhang University of California, Berkeley 
Curves on a smooth cubic surface in P3
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 HartshorneRao 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.

April 20  Christopher Eur University of California, Berkeley 
Divisors on matroids and their volumes
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 algebrogeometric and combinatorial properties in connection to graded linear series on blowups of projective spaces.

April 27  Linquan Ma University of Utah 
Perfectoid spaces

May 4  Kenta Sato University of Tokyo 
Ascending chain condition for Fpure thresholds
For a germ of a variety in positive characteristic and a nonzero ideal sheaf on the variety, we can define the Fpure 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 Fpure thresholds on a
fixed strongly Fregular 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.

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