On a conjecture of Lange

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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'<r, associate to E the invariant sᵣ'(E)=r'deg(E)-rdeg(E') where E'is a subbundle of E of rank r' and maximal degree. For every r', one can stratify the moduli space of stable vector bundles according to the value of the invariant. Lange's conjecture says that this strata are non-empty and of the right dimension if sᵣ'>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≤ 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.