Algebraic Geometry Seminar -- Fall 2013

This seminar meets 10:45 -- 11:45 Wednesday in room 312 LeConte.

Wednesday, September 11, 10:40 AM, room 312 LeConte

Speaker: Jesse Kass

Title: How does a n-by-(n+1) matrix deform?

Abstract: In his 1974 ICM lecture, V. I. Arnold outlined a program for studying a singularity defined by one equation -- a hypersurface singularity -- by deforming the equation, and over the past 39 years remarkable progress has been made on this program. One of the great achievements is the discovery that the simple hypersurface singularities are exactly the ADE singularities and that many features of these singularities can be described by the ADE Dynkin Diagrams.

What about more general classes of singularities? A natural class to consider are the codimension 2 singularities. Such a singularity can be given by equations that are the minors of a n-by-(n+1) matrix by a celebrated theorem of Burch-Hilbert-Schaps, and the singularity can be studied by deforming the matrix. In my talk, I will discuss extending parts of Arnold's program, especially the description of simple singularities, to codimension 2 singularities.

Fans of Graph theory: A collection of diagrams, analogous to the ADE Dynkin diagrams, naturally appears in this work. Have these collections appeared before in a different context?
Wednesday, September 18, 10:40 AM, room 312 LeConte:

Speaker: Matt Ballard

Title: Stratifications in Geometric Invariant Theory

Abstract: Given a group acting on a ring, the first question to ask is: what is the ring of invariants? In general, determining a presentation of the invariant ring, although a question of intense attention through classical times, is quite hard. If we table the question of explicit generators and relations and instead focus on the geometric process reflected in passing to invariant rings, we take a first step into Mumford's Geometric Invariant Theory (GIT).

Initially, GIT provided a nice way of forming "orbit spaces" in algebraic geometry. However, there is a certain non-uniqueness in forming GIT quotients and this non-uniqueness allows one to compare the geometry of GIT quotients arising from different choices.

In this talk, I will review some basics of GIT focusing on the structure of the unstable locus. This will lead into further, probably future, discussion of how different GIT quotients are related geometrically and algebraically. .
Wednesday, September 25, 10:45 AM, room 312 LeConte:

Speaker: Frank Thorne

Title: Prehomogeneous vector spaces and field extensions

Abstract: A representation (G, V) is prehomogeneous if there is a Zariski-open G(C) orbit in V(C). Typically one describes V as having a "singular set", so that G(C) acts transitively on the nonsingular set. These representations are also interesting over Q, Z, and other base rings. In this talk, I will describe work of Wright and Yukie, proving that many (G, V) parameterize field extensions over Q. I'll pay special attention to the case G = GL(2) x GL(3), V = Sym^2 C^3 (x) C^2, which parameterizes quartic fields. I will prove this, and I will explain how a special case was previously proved by Omar Khayyam in 1077, using algebraic geometry. I'll also say a little bit of the story over Z, and describe work of Bhargava as well as an open problem which I hope to solve.
Wednesday, October 2, 2013, 10:45 AM, room 312 Leconte:

Speaker: Adela Vraciu

Title: Examples of quasi-complete intersection ideals.

Abstract: Quasi-complete intersection ideals are ideals that have "exterior" Koszul homology. They include complete intersection ideals and share many of the homological change of rings properties of complete intersections. We will discuss several constructions and examples of quasi-complete intersections.
Wednesday, October 9, 2013, 10:45 AM, room 312 Leconte:

Speaker: Jim Coykendall (Clemson University).

Title: An Overview of Factorization: Algebraic and Graphical

Abstract: Since about 1990, there has been a large amount of effort devoted to the study of factorization in integral domains (as well as in other structures). Much of this study can be interpreted as an attempt to understand how the multiplicative structure of an integral domain "works" when we do not have unique factorization. A classical example is the class group, the size and complexity of which may be interpreted as a measure of "how far" a (Krull) domain is from being a UFD.
Wednesday, October 16, 2013, 10:45 AM, room 312 Leconte:

Speaker: Andy Kustin

Title: Blowups and fibers of morphisms

Abstract: We study the fibers of a rational map between two projective spaces from the point of view of Rees algebras.
Wednesday, October 23, 2013, 10:45 AM, room 312 Leconte:

Speaker: Matt Ballard

Title: Geometric Invariant Theory and derived categories

Abstract: I will remind everyone what the derived category is and then describe how different GIT quotients have closely related derived categories.
Wednesday, October 30, 2013, 10:45 AM, room 312 Leconte:

Speaker: Cameron Atkins

Title: Göbel's bound for the degrees of the generators of rings of invariants.
Monday, November 11, 2013, 10:45 AM, room 312 Leconte:

Speaker: Ken-ichiroh Kawasaki (Nara University of Education, Japan)

Title: Several results on characterizations of cofinite complexes.
Wednesday, December 4, 2013, 10:45 AM, room 312 Leconte:

Speaker: Michael Filaseta

Title: The genus behind Hilbert's Irreducibility Theorem

Abstract: Siegel showed that if f(x,y) in Z[x] is irreducible in Q[x,y], then there are finitely many integer points (x_{0},y_{0}) on the curve f(x,y) = 0 whenever the genus of the curve is positive. Observe that Siegel's result can be rephrased as saying that if the genus of the irreducible curve f(x,y) = 0 is positive, then there are at most finitely many integers y_{0} for which the polynomial f(x,y_{0}) in Z[x] has a monic linear factor in Z[x]. What about non-monic and non-linear factors, and what happens if the genus equals 0? Hilbert's Irreducibility Theorem supplies some information: if f(x,y) in Z[x] is irreducible in Q[x,y], then there are infinitely many y_{0} in Q such that f(x,y_{0}) is irreducible in Q[x]. In this talk, we will explore further connections between these two results. In particular, we discuss the question of whether Siegel's result implies Hilbert's Irreducibility Theorem, a question first addressed by Siegel himself.