On a conjecture of Lange

Let C be a projective smooth curve of genus g> 1. Let E be a vector bundle of rank r on C. For each integer r' 0. The conjecture has recently been solved thanks to work of Lange- Narasimhan, Lange-Brambila-Paz, Ballico and the authors. The purpose of this paper is to give a simpler proof of the result valid without further assumptions. The method of proof provides additional information on the geometry of the strata. We can prove that each strata (which is irreducible) is contained in the closure of the following one. We also show the unicity of the maximal subbundle when s\le r'(r-r')(g-1). Our methods can be used to study twisted Brill-Noether loci and to give a new proof of Hirschowitz Theorem about the non-speciality of the tensor product of generic vector bundles.