Lange's conjecture

In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by Herbert_Lange_(mathematician) and proved by Montserrat Teixidor i Bigas and Barbara Russo in 1999.

Statement
Let C be a smooth projective curve of genus greater or equal to 2. For generic vector bundles $$E_1$$ and $$E_2$$ on C of ranks and degrees $$(r_1, d_1)$$ and $$(r_2, d_2)$$, respectively, a generic extension
 * $$0 \to E_1 \to E \to E_2 \to 0$$

has E stable provided that $$\mu(E_1) < \mu(E_2)$$, where $$\mu(E_i) = d_i/r_i$$ is the slope of the respective bundle. The notion of a generic vector bundle here is a generic point in the moduli space of semistable vector bundles on C, and a generic extension is one that corresponds to a generic point in the vector space $\operatorname{Ext}^1$$$(E_2,E_1)$$.

An original formulation by Lange is that for a pair of integers $$(r_1, d_1)$$ and $$(r_2, d_2)$$ such that $$d_1/ r_1 < d_2/r_2$$, there exists a short exact sequence as above with E stable. This formulation is equivalent because the existence of a short exact sequence like that is an open condition on E in the moduli space of semistable vector bundles on C.