Speaker: Sameera Vemulapalli (Harvard University)

Title: Brill--Noether theory of smooth plane curves

Abstract:

Given a smooth curve C, it is natural to ask: what are all the degree $d$ maps from $C$ into a projective space $\mathbb{P}^r$? The study of this question is called Brill-Noether theory. Given a curve $C$, the data of a degree d map $C \rightarrow \mathbb{P}^r$ is equivalent to the data of a degree $d$ line bundle on $C$ together with a choice of $r + 1$ global sections having no common zeros. As such, a central object of study is the Brill–Noether locus $W^r_d(C)$, which is defined to be the space of degree $d$ line bundles on $C$ with at least $r+1$ global sections.

The famous Brill-Noether theorem gives a nice description of $W^r_d(C)$ when $C$ is a general curve of genus $g$. However, curves we come across in nature (such as curves in the plane) are not general, and may fail the Brill-Noether theorem! In this talk, I'll describe joint work with Hannah Larson, in which we describe the Brill-Noether theory of smooth plane curves (and more generally, curves on Hirzebruch surfaces), using tools from arithmetic statistics.