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Let X be a smooth projective curve of genus g bigger then 2. For any vectorbundle E on X let M_k(E) be the scheme of all rank k subbundles of E withmaximal degree. For every integers r, k and x with 0<k<r, x positive and eitherx less then (k-1)(r-2k+1) (if 2k is less then r) or (r-k-1)(2k-r+1) (if 2k> r),we construct a rank r stable vector bundle E such that M_k(E) has anirreducible component of dimension x. Furthermore, if there exists a stablevector bundle F with small Lange's invariant s_k(F) and with M_k(F) `spreadenough', then X i s a multiple covering of a curve of genus bigger then 2.

Families of Maximal Subbundles of Stable Vector Bundles on Curves