Vector Bundles and So(3)-invariants for Elliptic Surfaces Iii: the Case of Odd Fiber Degree

Let S be a simply connected elliptic surface with at most two multiple fibers, of multiplicities m1 and m2, where one or both of the mi are allowed to be 1. In this paper, the last of a series of three, we shall study stable rank two vector bundles V on S such that detV ·f is odd, where f is a general fiber of S. Thus necessarily the multiplicities m1 and m2 are odd as well. Bundles V such that det(V |f) has even degree for a general fiber f have been studied extensively [3], [4, Part II], and as we shall see the case of odd fiber degree is fundamentally different. Thus we shall have to develop the analysis of the relevant vector bundles from scratch, and the results in this paper are for the most part independent of those in [3] and [4]. Our goal in this paper is to give a description of the moduli space of stable rank two bundles with odd fiber degree and then to use this information to calculate certain Donaldson polynomials. Before stating our main result, recall that, for an elliptic surface S, J(S) denotes the elliptic surface whose general fiber is the set of line bundles of degree d on the general fiber of S. We shall prove the following two theorems: