Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces

Let X be a ruled surface over a nonsingular curve C of genus \(g\ge 0.\) 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.