# Algebraic Geometry Seminar Fall 2015

The seminar meets on Fridays at 2:25 pm in Van Vleck B223.

The schedule for the previous semester is here.

## Contents

## Algebraic Geometry Mailing List

- Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## Fall 2015 Schedule

date | speaker | title | host(s) |
---|---|---|---|

September 18 | Eric Riedl (UIC) | Rational Curves on Hypersurfaces | Jordan |

September 25 | David Zureick-Brown (Emory) | Hilbert schemes of canonically embedded curves of low genus | Jordan |

October 2 | Vasily Dolgushev (Temple) | A manifestation of the Grothendieck-Teichmueller group in geometry | Andrei |

October 9 | Laurentiu Maxim (Madison) | Equivariant invariants of external and symmetric products of quasi-projective varieties | local |

October 16 | Ed Dewey (Madison) | TBA | local |

October 23 | Jesse Kass (South Carolina) | How to count zeros arithmetically? | Melanie |

November 6 | Eric Ramos (Wisconsin) | TBA | Daniel |

November 13 | Xudong Zheng (UIC) | TBA | Daniel |

## Abstracts

### Eric Riedl

Rational Curves on Hypersurfaces

One way to understand the geometry of a variety is to understand its rational curves. Even for some relatively simple varieties, little is known about their spaces of rational curves. Many people have made previous progress on these questions, but there remain many open cases. In joint work with David Yang, we investigate the dimensions of the spaces of rational curves on very general hypersurfaces, and prove that for n > d+1 or d > (3n+1)/2, the spaces of rational curves have the expected dimension, as conjectured (in various cases) by several people, including Coskun, Harris and Starr, and Voisin. In this talk, we focus our attention particularly on the Fano case and try to motivate some of the ideas used to attack this problem.

### David Zureick-Brown

Hilbert schemes of canonically embedded curves of low genus

I'll discuss new work (joint with Aaron Landesman) on smoothability of low genus curves.

### Jess Kass

How to count zeros arithmetically?

A celebrated result of Eisenbud--Kimshaishvili--Levine computes the local Brouwer degree of a real polynomial function at an isolated zero as the signature of a quadratic form. I will discuss a parallel result in A1-homotopy theory, and time permitting, explain how to study a singularity by applying these results to the gradient of a defining equation. This is joint work with Kirsten Wickelgren.

### Vasily Dolgushev

A manifestation of the Grothendieck-Teichmueller group in geometry

Inspired by Grothendieck’s lego-game, Vladimir Drinfeld introduced, in 1990, the Grothendieck-Teichmueller group GRT. This group has interesting links to the absolute Galois group of rationals, moduli of algebraic curves, solutions of the Kashiwara-Vergne problem, and theory of motives. My talk will be devoted to the manifestation of GRT in the extended moduli of algebraic varieties, which was conjectured by Maxim Kontsevich in 1999. My talk is partially based on the joint paper with Chris Rogers and Thomas Willwacher: http://arxiv.org/abs/1211.4230.

### Laurentiu Maxim

Equivariant invariants of external and symmetric products of quasi-projective varieties

I will start by revisiting formulae for the generating series of genera of symmetric products (with suitable coefficients), which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical results in the literature as special cases. Important specializations of these results include generating series for extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures of symmetric products of complex projective varieties. In the second part of the talk, I will describe a generating series formula for equivariant invariants of external products, which includes all of the above-mentioned results as special cases. This is joint work with Joerg Schuermann.