Flexible varieties and automorphism groups

Abstract

Abstract. Given an irreducible affine algebraic variety X of dimension n ≥ 2, we let SAut(X) denote the special automorphism group of X i.e., the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x ∈ X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We provide also various modifications and applications.

Flexible affine cones over del Pezzo surfaces of degree 4

“…In addition, [1] proves that every flexible variety is unirational. On the other hand, in [4], it is conjectured that every unirational variety is stably birational to an infinitely transitive variety and it is proved in some cases.…”

“…Moreover, in [3], it is proved that the suspensions over flexible varieties are also flexible. One can find some other examples of flexible varieties in [1,2].…”

“…Thus all A 1 -fibers of the cylinder U 2 intersect L, so that all of them should be contained in W . 1 Therefore, the complement S\W is contained in the support of finitely many curves.…”

On rigidity of factorial trinomial hypersurfaces

“…It is easy to deduce from this description that every affine toric variety different from a torus admits a non-trivial G a -action. Moreover, by [4, Theorem 2.1] every nondegenerate affine toric variety of dimension at least 2 is flexible in the sense of [2]. In particular, it admits many G a -actions.…”

Гибкость Аффинных Конусов Над Поверхностями Дель Пеццо Степени 4 И 5

“…Аффинное алгебраическое многообразие X , определенное над алгебраически замкнутым полем K характеристики нуль, называется гибким, если касательное пространство к X в произвольной гладкой точке порождается касательными векторами к орбитам действия однопараметрических унипотентных групп [3]. В этой работе мы установим гибкость аффинных конусов над поверхностями дель Пеццо степени 4 и 5.…”