Abstract
Let X be a smooth projective curve of genus g ≥ 2, and let E be a vector bundle on X. Let M k (E) be the scheme of all rank k subbundles of E with maximal degree. For every integer r, k and x with 0 < k < r and either 2k ≤ r and 0 ≤ x ≤ (k − 1)(r − 2k + 1) or 2k > r and 0 ≤ x ≤ (r − k − 1)(2k − r + 1), we construct a rank r stable vector bundle E such that M k (E) has an irreducible component of dimension x. Furthermore, if there exists a stable vector bundle F with small Lange's invariant s k (F) and with M k (F) 'spread enough,' then X is a multiple covering of a curve of genus bigger than 2. 1. Introduction. Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field K. In this paper we study the rank r stable vector bundles, E, on X such that for some integer k with 0 < k < r, E has a 'large' family of subbundles with rank k and maximal degree. For positive integers r, d let M (X; r, d) be the moduli space of stable vector bundles on X of rank r and degree d. It is well known that M (X; r, d) is smooth and irreducible. For a positive integer k with 0 < k < r, let M k (E) be the set of all rank k subbundles of E with maximal degree. Being a Quot-scheme, M k (E) has a natural scheme-structure. For the intent of this paper we will only need to consider its reduced structure. Indeed, we are interested in finding a stable vector bundle E such that M k (E) has an irreducible component with prescribed dimension. Since every element in M k (E) has maximal degree, the scheme M k (E) is complete. Hence, by [7, pp. 254 255], we have dim (M k (E)) ≤ k(r − k) for every rank r vector bundle E. Fixing x with x ≤ k(r − k), it is very easy to find a decomposable rank r vector bundle E such that M k (E) has an irreducible component of dimension x. But we are interested in stable vector bundles which are indecomposable. Hence, using extensions of