Lange's conjecture
In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by 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 formula_1 and formula_2 on "C" of ranks and degrees formula_3 and formula_4, respectively, a generic extension has "E" stable provided that formula_6, where formula_7 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 formula_8formula_9. An original formulation by Lange is that for a pair of integers formula_3 and formula_4 such that formula_12, 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".