Back to Algebraic Geometry Seminar: Spring 2014
Algebraic Geometry Seminar
Date: Tuesday, Mar 18, 2014
Time: 3:30PM - 4:30PM
Location: LCB 215
Brian Osserman
UC Davis
Title |
Recent progress on vector bundles with sections |
Abstract |
Higher-rank Brill-Noether, at its most basic, seeks to answer the following question: how many global sections can a (semistable) vector bundle of given rank and degree have on general curve of genus g? When there exist vector bundles with a k-dimensional space of sections, one then asks how many such bundles there are. The classical rank-1 version was settled completely in the 1970’s and 1980’s, but even the rank-2 case is still wide open, with many partial results but no comprehensive conjecture. In the 1990’s, Bertram, Feinberg and Mukai observed that the canonical determinant case has special behavior, with a higher expected dimension, and I will discuss recent work which studies the special determinant case more systematically. This work is in two parts: producing new expected dimensions on smooth curves by exploiting symmetries to produce lower bounds on dimensions, and developing the general theory of higher-rank limit linear series in order to use degeneration arguments to prove existence results. Some of this is joint work with Montserrat Teixidor i Bigas. |
Algebraic Geometry