## Algebra & Number Theory

### Del Pezzo surfaces and representation theory

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 1, Number 4 (2007), 393-419.

Dates
Revised: 11 August 2007
Accepted: 15 September 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797169

Digital Object Identifier
doi:10.2140/ant.2007.1.393

Mathematical Reviews number (MathSciNet)
MR2368955

Zentralblatt MATH identifier
1170.14026

#### Citation

Serganova, Vera; Skorobogatov, Alexei. Del Pezzo surfaces and representation theory. Algebra Number Theory 1 (2007), no. 4, 393--419. doi:10.2140/ant.2007.1.393. https://projecteuclid.org/euclid.ant/1513797169

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