Irreducibility of Moduli Spaces of Vector Bundles on Birationally Ruled Surfaces

Let S be a birationally ruled surface. We show that the moduli schemes MS(r, c1, c2) of semistable sheaves on S of rank r and Chern classes c1 and c2 are irreducible for all (r, c1, c2) provided the polarization of S used satisfies a simple numerical condition. This is accomplished by proving that the stacks of prioritary sheaves on S of fixed rank and Chern classes are smooth and irreducible. One important recent result in the theory of vector bundles on algebraic surfaces is the theorem of Gieseker and Li that for any smooth projective surface S and any ample divisor H on S, the moduli scheme MS,H(2, c1, c2) of S-equivalence classes of H-semistable torsion-free sheaves of rank 2, determinant c1 ∈ Pic(S), and second Chern class c2 is irreducible if c2 ≫ 0. If S is a surface of general type, the condition c2 ≫ 0 is necessary because of an example of Gieseker with small c2 where the moduli space is reducible. In contrast it has been known for quite some time that the moduli schemes MP2,H(r, c1, c2) is irreducible for all (r, c1, c2) for which there exist semistable sheaves on the projective plane, and the same result is also known for P1 × P1. In this paper we extend this strong irreducibility result from P2 and P1 × P1 to all smooth projective surfaces of negative Kodaira dimension. To simplify our exposition, we will omit P2 although it can be handled by the same method. (Indeed our method is based on a method of Ellingsrud and Strømme which was developed for P2.) So our surface S possesses a morphism π: S → C onto a smooth curve with connected fibers and with general fiber isomorphic to P1. We fix such a π. (Such a π is unique if q(S) = g(C) > 0 or if S = P(OP1 ⊕ OP1(e)) with e > 0, but there can be many, even infinitely many, possible π for certain rational surfaces.) For p ∈ C, let fp = π −1(p). These fp are all numerically equivalent, and we write f ∈ NS(X) for the numerical class of these fp. We prove Theorem 1. Let π: S → C be a birationally ruled surface and f ∈ NS(S) the numerical class of a fiber of π. Let H be an ample divisor on S such that H · (KS + f) < 0. Suppose r ≥ 2, c1 ∈ NS(S), and c2 ∈ Z are given. If the moduli scheme MS,H(r, c1, c2) of S-equivalence ∗Supported in part by NSA research grant MDA904-92-H-3009.