Lange's conjecture explained

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

E _{1} and E _{2} on C of ranks and degrees (r _{1,} d _{1)} and (r _{2,} d _{2)} , respectively, a generic extension0 \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.

References

• Lange . Herbert . Zur Klassifikation von Regelmannigfaltigkeiten . 10.1007/BF01456060 . 696517 . 1983 . . 0025-5831 . 262 . 4 . 447–459.
• Teixidor i Bigas. Montserrat. Montserrat Teixidor i Bigas. Barbara. Russo. On a conjecture of Lange . alg-geom/9710019 . 1689352 . 1999 . Journal of Algebraic Geometry . 1056-3911 . 8 . 3 . 483–496. 1997alg.geom.10019R .
• Ballico. Edoardo . Extensions of stable vector bundles on smooth curves: Lange's conjecture . 2000 . Analele Ştiinţifice ale Universităţii "Al. I. Cuza" din Iaşi. (N.S.) . 46 . 1 . 149–156 . 1840133.