Brill-Noether Theory of stable vector bundles on ruled surfaces

Let X be a ruled surface over a nonsingular curve C of genus $g\geq0$. Let $M_H:=M_{X,H}(2;c_1,c_2)$ 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 $W_H^k(2;c_1,c_2)$, 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 $W_H^k(2;c_1,c_2)$ is non-empty.