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Lange's conjecture

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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.

Statement

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Let C be a smooth projective curve of genus greater or equal to 2. For generic vector bundles and on C of ranks and degrees and , respectively, a generic extension

has E stable provided that , where 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 .

An original formulation by Lange is that for a pair of integers and such that , 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

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  • Lange, Herbert (1983). "Zur Klassifikation von Regelmannigfaltigkeiten". Mathematische Annalen. 262 (4): 447–459. doi:10.1007/BF01456060. ISSN 0025-5831. MR 0696517.
  • Teixidor i Bigas, Montserrat; Russo, Barbara (1999). "On a conjecture of Lange". Journal of Algebraic Geometry. 8 (3): 483–496. arXiv:alg-geom/9710019. Bibcode:1997alg.geom.10019R. ISSN 1056-3911. MR 1689352.
  • Ballico, Edoardo (2000). "Extensions of stable vector bundles on smooth curves: Lange's conjecture". Analele Ştiinţifice ale Universităţii "Al. I. Cuza" din Iaşi. (N.S.). 46 (1): 149–156. MR 1840133.

Notes

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  1. ^ Herbert Lange (1983)