Algebraic Geometry Progress Seminar
Fall 2009

This informal seminar is organized by Melanie Matchett Wood (with the collaboration of Ravi Vakil).
We'll have informal talks, in sequences of two, in which people discuss things related to their work in a down-to-earth manner intended for a broad audience.
The seminar meets Mondays 2:30-3:30 pm, in 383-N.

Informal Lunch: Also, some algebraic geometers meet informally for lunch (bring your own) from 12-1 on Wednesday's in the common room on the second floor.  Everyone is welcome, including students and those with tangential interests in algebraic geometry.

Schedule of Talks

September 28 and October 12: Christian Liedtke, Degenerations of elliptic curves, Neron models and the Tate algorithm (Reference: Chapter IV of Silverman's book "Advanced topics in the arithmetic of elliptic curves")

Special Seminar OCTOBER 12, 5PM room 383-N: Joe Rabinoff (Harvard): A brief introduction to tropical geometry and an application to arithmetic geometry
Special Algebraic Geometry dinner, following the talk

October 5 and October 19: Anthony Bak, Geometric Langlands for GL(1)

October 26 and November 2: Andrew Snowden: Projective embeddings of spaces of points

November 9 and November 16: Dimitri Zvonkine: Open problems in the intersection theory of moduli spaces of curves

November 23 no meeting (Thanksgiving week)

November 30 and December 7

Abstracts
Christian Liedtke, Degenerations of elliptic curves, Neron models and the Tate algorithm

The idea is to give some motivation of what one is looking for in the theory of Neron models, i.e., finding good models for degenerations of Abelian varieties where having a group structure is the guideline rather than have only "few" singularities in the bad fibers. Next, I will explain how to obtain the degeneration types for elliptic curves, i.e., the Kodaira-Neron classification via extended Dynkin diagrams. Then, I will explain how to obtain these types from a given Weierstrass equation via the Tate algorithm, which is extremely explicit. Finally, I will give some applications of this explicit algorithm to more sophisticated topics, such as Galois actions on Tate modules, semi-stable reduction, Ogg's formula, Swan conductors.

Anthony Bak: Geometric Langlands for GL(1)

Abstract: We will motivate the geometric Langlands conjecture in a
purely geometric context by considering it as some kind of "Fourier
transform for reductive groups over curves".  We will consider three
different proofs (including an analytic one) of the Gl(1) case and
discuss some connections with the general conjecture.

Joe Rabinoff (Harvard): A brief introduction to tropical geometry and an application to arithmetic geometry

Abstract:
Tropical geometry would perhaps be better named non-Archimedean algebraic geometry, as it studies the valuations of the zeros of polynomial equations over non-Archimedean fields.  I'll give a concise introduction to tropical geometry, and then give an idea of how it can be used to calculate the number of zeros of a set of polynomial equations (or power series!) whose coordinates have a given valuation.  This is direct generalization of the theory of the Newton polygon to higher dimensions, and has proved very useful in arithmetic geometry.

Andrew Snowden: Projective embeddings of spaces of points

Abstract:
Let X be a projective variety with an action of a reductive group G.  The subject of my two talks will be the projective coordinate ring of the GIT quotient X^n/G, especially its behavior as a function of n.  In the first talk, I'll discuss the most basic case where X=P^1 and G=GL(2).  In this case, the coordinate ring is generated in degree 1 for all (even) n and the ideal of relations between these generators is generated by quadratics (for n not 6).  (The first result is due to Kempe, from the 19th century; the second is a recent result from my collaboration with Howard, Millson and Vakil.)  In the second talk, I'll discuss the general case and formulate some conjectures that would generalize the results from the P^1 case.

Dimitri Zvonkine: Open problems in the intersection theory of moduli spaces of curves

Abstract:
Since not much is known about the cohomology ring of  M_{g,n}, it seems to be a general consensus to study its subring called the tautological ring. Although for large  g  and  n  the tautological ring is known to be much smaller than the full cohomology ring, all geometric classes one can think of usually belong to this subring. Here are the three problems that I will discuss.

1. Describe the tautological ring of  M_{g,n}.  For  n=0  a conjectural description was given by Carel Faber in 1993 and is still not proved to be correct. I now have a similar conjectural description for all  M_{g,n}.  Proving these conjectures can be considered a combinatorial problem; ideed there is an algorithm, that I will describe, that proves the conjectures for any given  g  and  n granted sufficient computation power.

2. On a genus 0  curve there exists a function with any prescribed zeroes and poles. This is no longer true in higher genus. The condition of the existence of a meromorphic function with given zeroes and poles cuts out a cycle in the moduli space of curves with marked points. Finding the cohomology class PoicarĂ© dual to this cycle is a very important problem with several applications.

3. An ELSV-type formula for the space of  r-spin structures. The ELSV formula relates Hurwitz numbers (that count ramified coverings of  CP^1)  to intersection numbers on  Mbar_{g,n}.  It is one of the most important results in
the intersection theory of moduli spaces. I will present a conjectural generalization of this formula to the space of  r-spin structures (curves with an  r-th tensor root of the cotangent line bundle).