Del Pezzo surfaces and representation theory

Details Del Pezzo surfaces and representation theory Del Pezzo surfaces and representation theory

2006-11-23

arxiv.org/abs/math/0611737v2

J. Algebra and Number Theory 1 (2007) 393-419.

Mathematics

Algebraic Geometry

revised version, 26 pages, to appear in J. Algebra and Number Theory

Scientific paper

In his book "Cubic forms" Manin discovered that del Pezzo surfaces are related to root systems. To explain the many numerical coincidences Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded in a certain projective homogeneous space of the semisimple group with the same root system, equivariantly with respect to the maximal torus action. We prove this conjecture for del Pezzo surfaces of degrees greater than 1. Our proof uses an inductive process based on representations of Lie algebras corresponding to Hermitian symmetric pairs.

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