Geography of Brill-Noether loci for small slopes

TL;DR

This paper investigates the geometric properties of Brill-Noether loci for stable bundles on algebraic curves, establishing conditions for their irreducibility, smoothness, and non-emptiness in the case of small slopes.

Contribution

It proves irreducibility, smoothness, and non-emptiness of Brill-Noether loci for small slopes, providing new criteria and extending understanding of their geometric structure.

Findings

Brill-Noether loci are irreducible and smooth outside the loci when non-empty.

Non-emptiness occurs if and only if specific numerical conditions are met.

Irreducibility and non-emptiness are established for semistable loci.

Abstract

Let $X$ be a non-singular projective curve of genus $g\ge2$ over an algebraically closed field of characteristic zero. Let $\mo$ denote the moduli space of stable bundles of rank $n$ and degree $d$ on $X$ and $\wn $ the Brill-Noether loci in $\mo .$ We prove that, if $0\leq d \leq n $ and $\wn $ is non-empty, then it is irreducible of the expected dimension and smooth outside $\wnn$. We prove further that in this range $\wn$ is non-empty if and only if $d>0$, $n\leq d+(n-k)g$ and $(n,d,k) \not= (n,n,n)$. We also prove irreducibility and non-emptiness for the semistable Brill-Noether loci.