Families of Maximal Subbundles of Stable Vector Bundles on Curves

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 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.