On stable and ample vector bundles of rank 2 on curves

Abstract

IrE is a vector bundle of rank 2 on a curve X over an algebraically closed field k and f: Y--,X a finite covering, the sub line bundles of E yield sub line bundles of f*E . But apart from those there may be some "new" ones in f*E in a sense to be made precise (cp. Definition (1.1)). It will be shown (cp. Theorem (1.2)) that for a fixed Y and all finite coverings f : Y--,X of degree n these correspond in a natural and bijective way to the integral curves D on the projective bundle P(E) of degree n overX such that the normalization o lD is isomorphic to Y. This is almost trivial in characteristic 0 but not so easy in characteristic >0 because of the existence of isomorphisms F*E~-E®L where F denotes the Frobenius morphism and L a suitable line bundle on X. Using intersection calculus on the surface P(E) it turns out that the degree of such a "new" sub line bundle of f*E is uniquely determined by E, f and the arithmetical genus of the corresponding curve on P(E) and can in fact be computed (cp. Theorem (1.3)). This allows to study the behaviour of the stability degree of these vector bundles under pull backs by finite coverings (cp. Sects. 2 and 3) and at last to give the corresponding result for positive characteristics (cp. Proposition (4.4)) of Hartshornes characterization of ample vector bundles of rank 2 on curves in characteristic 0 (cp. [4], (7.6)). In the whole of the paper we suppose without further stating for the genus of X : g(X) > 2. For 9(x) = 0 and 1 the results of Sects. 2, 3, and 4 are well known. On the other hand it is not difficult to prove the corresponding results to Theorems (1.2) and (1.3) for g(X)=0 and 1. I would like to thank W. Barth and W. D. Geyer for pointing out a mistake in a preliminary version of this paper.