Brill-Noether theory of stable vector bundles on ruled surfaces

Abstract

Let $X$ be a ruled surface over a nonsingular curve $C$ of genus $g \geq 0$. Let $M_H:=M_{X, H}\left(2 ; c_1, c_2\right)$ be the moduli space of $H$-stable rank 2 vector bundles $E$ on $X$ with fixed Chern classes $c_i:=c_i(E)$ for $i=1,2$. The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space $M_H$ in terms of its Brill-Noether locus $W_H^k\left(2 ; c_1, c_2\right)$, whose points correspond to stable vector bundles in $M_H$ having at least $k$ independent sections. We deal with the non-emptiness of this Brill-Noether locus, getting in most of the cases sharp bounds for the values of $k$ such that $W_H^k\left(2 ; c_1, c_2\right)$ is non-empty.