Del Pezzo surfaces and representation theory

Serganova, Vera; Skorobogatov, Alexei

doi:10.2140/ant.2007.1.393

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Del Pezzo surfaces and representation theory

Vera V. Serganova and Alexei N. Skorobogatov

Vol. 1 (2007), No. 4, 393–419

DOI: 10.2140/ant.2007.1.393

Abstract

The connection between del Pezzo surfaces and root systems goes back to Coxeter and Du Val, and was given modern treatment by Manin in his seminal book Cubic forms. Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded into a certain projective homogeneous space of the semisimple group with the same root system, equivariantly with respect to the maximal torus action. Computational proofs of this conjecture based on the structure of the Cox ring have been given recently by Popov and Derenthal. We give a new proof of Batyrev’s conjecture using an inductive process, interpreting the blowing-up of a point on a del Pezzo surface in terms of representations of Lie algebras corresponding to Hermitian symmetric pairs.

To Yuri Ivanovich Manin on his seventieth birthday

Keywords

del Pezzo surface, homogeneous space, Lie algebra

Mathematical Subject Classification 2000

Primary: 14J26

Secondary: 17B25, 17B10

Milestones

Received: 2 February 2007

Revised: 11 August 2007

Accepted: 15 September 2007

Published: 1 November 2007

Authors

Vera V. Serganova
Department of Mathematics University of California Berkeley, CA 94720-3840 United States

Alexei N. Skorobogatov
Department of Mathematics South Kensington Campus Imperial College London SW7 2BZ United Kingdom	Institute for Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetnyi Moscow, 127994 Russia

Vol. 1, No. 4, 2007