Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces

Abstract

Let X be a ruled surface over a nonsingular curve C of genus $$g\ge 0.$$ g ≥ 0 . Let $$M_H:=M_{X,H}(2;c_1,c_2)$$ 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 $$c_i:=c_i(E)$$ c i : = c i ( E ) for $$i=1,2.$$ i = 1 , 2 . The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space $$M_H$$ M H in terms of its Brill–Noether locus $$W_H^k(2;c_1,c_2),$$ W H k ( 2 ; c 1 , c 2 ) , whose points correspond to stable vector bundles in $$M_H$$ 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 $$W_H^k(2;c_1,c_2)$$ W H k ( 2 ; c 1 , c 2 ) is non-empty.