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

Let S be a simply connected elliptic surface with at most two multiple fibers. In this paper, the second in a series of three, we are concerned with describing moduli spaces of stable vector bundles V over S such that the restriction of c1(V ) to a general fiber has the smallest possible nonzero degree, namely the product of the multiplicities, in the case where this product is even. We then apply this study toward a partial calculation of the corresponding Donaldson polynomial invariants of S. Our goal is the completion of the C classification of such surfaces, and the general outline of this classification has been described in the introduction to Part I. Aside from quoting a few results from Part I, this paper can however be read independently. On the other hand, the methods of this paper draw heavily on the book [4], and many arguments which are very similar to arguments in [4] are sketched or simply omitted. Roughly speaking, the new ingredients in the proof consist of the algebraic geometry of certain elliptic surfaces associated to S, which have a single multiple fiber of multiplicity two and are birational to double covers of rational ruled surfaces. The vector bundle parts of the argument run more or less parallel to the arguments in [4], with a few new cases to analyze. The outline of this paper is as follows. In this paper, we shall only be concerned with elliptic surfaces S over P with multiple fibers of multiplicities 2m1 and m2, where m2 is odd, and such that there exists a divisor ∆ on S with ∆ · f = 2m1m2, the minimum possible value, for a smooth fiber f . In this case, there is an associated surface J12(S) defined in [3]. The surface J12(S) fibers over P and the fiber over a point t lying under a smooth fiber f of S is J12(f), the set of line bundles of degree m1m2 on the fiber f of S. The surface J 12(S) has an involution defined by λ ∈ J12(f) 7→ Of (∆|f) ⊗ λ. The quotient of J12(S) by this involution is birational to a rational ruled surface FN , and we describe the geometry of the double cover in detail. In Section 2, we describe the rough classification of stable bundles V on S with c1(V ) = ∆. To each such bundle there is an associated bisection C of J12(S) which is invariant under the involution, and so defines a section of the quotient ruled surface. In Section 3, we show that for general bundles V , V is determined up to finite ambiguity by the section of the ruled surface and the choice of a certain line bundle on the associated bisection C of J12(S).