Lange's conjecture

In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by {{ill|Herbert_Lange_(mathematician)|lt=Herbet Lange|de|Herbert_Lange_(Mathematiker)|vertical-align=sup}}{{harvs|txt|last=Lange|first=Herbert|year=1983}} 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.

References

{{cite journal | last=Lange | first=Herbert | title=Zur Klassifikation von Regelmannigfaltigkeiten | doi=10.1007/BF01456060 |mr=696517 | year=1983 | journal=Mathematische Annalen | issn=0025-5831 | volume=262 | issue=4 | pages=447–459}}
{{cite journal |last1=Teixidor i Bigas|first1=Montserrat|author1-link= Montserrat Teixidor i Bigas|first2=Barbara|last2= Russo| title=On a conjecture of Lange | arxiv=alg-geom/9710019 |mr=1689352 | year=1999 | journal=Journal of Algebraic Geometry | issn=1056-3911 | volume=8 | issue=3 | pages=483–496| bibcode=1997alg.geom.10019R }}
{{cite journal | last= Ballico|first=Edoardo | title = Extensions of stable vector bundles on smooth curves: Lange's conjecture | year = 2000 | journal =

Analele Ştiinţifice ale Universităţii "Al. I. Cuza" din Iaşi|series= (N.S.) | volume=46 | issue = 1 | pages = 149–156 | mr=1840133}}

=Notes=

Category:Vector bundles

Category:Algebraic curves

Category:Theorems in algebraic geometry

Category:Conjectures that have been proved