Universal torsors of Del Pezzo surfaces and homogeneous spaces

摘要

Let Cox(Sr) be the homogeneous coordinate ring of the blow-up Sr of in r general points, i.e., a smooth Del Pezzo surface of degree 9?r. We prove that for r{6,7}, Proj(Cox(Sr)) can be embedded into Gr/Pr, where Gr is an algebraic group with root system given by the primitive Picard lattice of Sr and PrGr is a certain maximal parabolic subgroup.