  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
s_{r'}(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 s_{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.

Algebraic Geometry;Algebraic Geometry