On a Conjecture of Lange

We want to study this stratification. A vector bundle E ∈ Ur′,s(r, d) can be written in an exact sequence 0 → E ′ → E → E ′′ → 0 with E , E ′′ vector bundles of ranks r, r and degrees d, d satisfying r = r+r, d = d + d, rd − rd = rd − rd = s. Note that the condition s > 0 is equivalent to the inequality of slopes μ(E ) < μ(E ). One expects that when this condition is satisfied, a generic such extension will yield a stable E. We call this statement Lange’s conjecture (cf. [L]). The conjecture is now solved and a great deal is known about the geometry of the strata : the rank two case is treated in [L,N], the case s ≤ min(r, r)(g − 1) in [B,B,R]. In [T1], the result is proved for the generic curve and for every curve if E is assumed to be only semistable. This apparently implies the result also for E stable (cf.[B]). In [B,L], a proof is provided for g ≥ (r + 1)/2. The purpose of this paper is to give a simpler proof of the result valid without further assumptions. The method is somehow the converse of the one used by Brambila-Paz and Lange in [B,L]. They start with the most general E in U(r, d) and then show that a suitable transformation of E gives a new bundle with smaller s. They need to check then that such an E is in fact stable. Here, we start with an E with the smallest possible s and produce an E ′ with larger s. Stability then