Lange's conjecture
# Lange's conjecture

In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by Herbet Lange 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.

