Components of the Stack of Torsion-Free Sheaves of Rank 2 on Ruled Surfaces

Let S be a ruled surface without sections of negative self-intersection. We classify the irreducible components of the moduli stack of torsion-free sheaves of rank 2 sheaves on S. We also classify the irreducible components of the Brill-Noether loci in Hilb (P × P) given by W 0 N (D) = {[X ] | h1(IX(D)) ≥ 1} for D an effective divisor class. Our methods are also applicable to P giving new proofs of theorems of Strømme (slightly extended) and Coppo. Let π: S = P(A) → C be a ruled surface with tautological line bundle O(1) := OP(A)(1). The current classification of isomorphism classes of rank 2 vector bundles E on S ([BS] [Br] [HS] [Ho]) proceeds by stratifying the moduli functor (or stack) and then classifying the sheaves in each stratum independently. The numerical data used to distinguish the strata are usually (i) the splitting type OP1(a)⊕OP1(b) of the generic fiber of π (with a ≥ b), and (ii) the degree of the locally free sheaf π∗(E(−a)) on C. On each stratum, U := π (π∗(E(−a)))(a) is naturally a subsheaf of E , and the possible quotient sheaves E/U and extension classes Ext(E/U ,U) have been classified. To the author’s knowledge, rank 2 torsion-free sheaves on S have not been given a similar classification, but one could clearly adapt the ideas used for vector bundles. What this approach has usually not described is the relationship between the strata particularly for the strata parametrizing only unstable sheaves. In this paper we give a first result along these lines by describing which strata are generic, i.e. which are open in the (reduced) moduli stack. Thus we are really classifying the irreducible components of the moduli stack of rank 2 torsion-free sheaves on S. We use a method developed by Strømme [S] for rank 2 vector bundles on P2 modified by deformation theory techniques which originate in [DL]. We will divide our irreducible components into two types. The first type we call prioritary because the general member of a component of this type is a prioritary sheaf in the sense that we used in [W1]. That is, if for each p ∈ C we write fp := π −1(p) for the corresponding fiber, ∗Supported in part by NSA research grant MDA904-92-H-3009.