Vol. 1, No. 4, 2007

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Del Pezzo surfaces and representation theory

Vera V. Serganova and Alexei N. Skorobogatov

Vol. 1 (2007), No. 4, 393–419

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

del Pezzo surface, homogeneous space, Lie algebra
Mathematical Subject Classification 2000
Primary: 14J26
Secondary: 17B25, 17B10
Received: 2 February 2007
Revised: 11 August 2007
Accepted: 15 September 2007
Published: 1 November 2007
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