论文部分内容阅读
We study two families of affine verieties covered by toric open charts and prove their flexibility.Let X be an affine algebraic variety of dimension ≥ 2 defined over an algebraically closed field K of characteristic zero,and let SAut be the subgroup of automorphism group AutX that is generated by the 1-parameter unipotent subgroups,i.e.actions of the additive group Ga = Ga(K).A variety X is called flexible if the tangent space to X at an arbitrary regular point x ∈ X is generated by tangent vectors to orbits of Ga-actions.This is equivalent to the infinite transitivity of the action of SAut on the regular locus Xreg(∈)X,see [AFKKZ].Previously described classes of flexible varieties include: affine cones over flag varieties,non-degenerate toric varieties of dimension ≥2,and suspensions over flexible varieties [AKZ]; affine cones over del Pezzo surfaces of degree ≥4 [P]; universal torsors over A-covered varieties [APS].We will prove flexibility of the following families of varieties:(a)affine cones over secant varieties of Veronese–Segre varieties;(b)total coordinate spaces of smooth projective T-varieties of complexity 1.We use the construction from [KPZ] that provides a certain correspondence between open cylindric subsets on a projective variety Y and regular Ga-actions on the affine cone over Y.The talk is based on the joint work with Hendrik Süβ and Mateusz Michalek,and is partially supported by SPbGU grant for postdocs no.6.50.22.2014.