In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by Herbet Lange [de][1] and proved by Montserrat Teixidor i Bigas and Barbara Russo in 1999.
Let C be a smooth projective curve of genus greater or equal to 2.
For generic vector bundles
on C of ranks and degrees
r
, respectively, a generic extension has E stable provided that
μ (
) < μ (
μ (
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
Ext
{\displaystyle \operatorname {Ext} ^{1}}
An original formulation by Lange is that for a pair of integers
, 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.
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