# Difference between revisions of "Algebraic Geometry Seminar Fall 2015"

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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. | 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=== | ===Jess Kass=== |

## Revision as of 13:39, 23 September 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) | TBA | local |

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

November 13 | Jake Levinson (Michigan) | 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.