c © 1998 Universitat de Barcelona The subbundles of decomposable vector bundles over an elliptic curve

Let C be an elliptic curve and E, F polystable vector bundles on C such that no two among the indecomposable factors of E ⊕ F are isomorphic. Here we give a complete classification of such pairs (E,F) such that E is a subbundle of F. In [1] M. Atiyah classified the vector bundles over an elliptic curve C. But still, there are several natural open questions on the structure of the subbundles of a fixed vector bundle on C. The aim of this paper is to give a reasonably complete answer to this question restricting slightly the vector bundles involved (see Corollary 0.2). A vector bundle F on a smooth projective curve is called polystable if it is the direct sum of stable vector bundles with the same slope µ(F). In particular a polystable vector bundle is semistable. The notion of polystability is very natural over an elliptic curve, because very few vector bundles over an elliptic curve are stable, while for all integers r, d with r> 0 there exist polystable vector bundles with rank r and degree d. All this paper is devoted to the proof of the following result. Theorem 0.1 Fix integers x, y, ai, 1 ≤ i ≤ x, ri, 1 ≤ i ≤ x, bj, 1 ≤ j ≤ y, sj, 1 ≤ j ≤ y, with x> 0, y> 0, ri> 0 for every i, sj> 0 for every j and ai/ri < bj/sj for every i and every j. Let C be an elliptic curve. Fix polystable vector bundles Ei, 1 ≤ i ≤ x, and Fj, 1 ≤ j ≤ y, with rank(Ei) = ri, deg(Ei) = ai, rank(Fj) = sj