Jack Huizenga : Brill-Noether theory for vector bundles on the projective plane

Posted by Izzet Coskun , part of the Algebraic Geometry Seminar.

At

Sept. 15, 2025, 3 p.m.

In

636 SEO

Abstract

The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for vector bundles and higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. In this talk, we will study Brill-Noether loci for vector bundles on the projective plane in the case where the number of sections is close to the largest possible number. When the number of sections is very large, Brill-Noether problems are all "trivial"--the Brill-Noether loci are either empty or the entire moduli space. As the number of sections decreases, we find that there is a "first" nontrivial Brill-Noether locus, and we discuss its geometry.