  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.
