Martens and Mumford theorems for higher rank Brill--Noether loci

Abstract

Generalizing the Martens theorem for line bundles over a curve $C$, we obtain upper bounds on the dimension of the Brill--Noether locus $B_{n,d}^k$ parametrizing stable bundles of rank $n\ge 2$ and degree $d$ over $C$ with at least $k$ independent sections. This proves a conjecture of the second author and generalizes bounds obtained by him in the rank two case. We give more refined results for some values of $d$, including a generalized Mumford theorem for $n\ge 2$ when $d\le g-1$. The statements are obtained chiefly by analysis of the tangent spaces of $B_{n,d}^k$. As an application, we show that for $n\ge 5$ the locus $B_{n,n(g-1)}^2$ is irreducible and reduced for any $C$.

10 pages