Infinite transitivity on universal torsors

Abstract

Let X be an algebraic variety covered by open charts isomorphic to the affine space and q : X → X be the universal torsor over X. We prove that the special automorphism group of the quasi-affine variety X acts on X infinitely transitively. Also we find wide classes of varieties X admitting such a covering.

Flexible affine cones and flexible coverings

“…0.2] for the case of degree 5, where the total coordinate space is known to be the affine cone over the Grassmannian G (2,5). On the other hand, this extends a result of Arzhantsev et al [6], where flexibility was proved only outside a subset of codimension 2. …”

“…In Arzhantsev et al [6] it was shown that smooth varieties of this type admit a toric covering and for certain affine cones over these varieties we, indeed, obtain flexibility. For example, this applies to all known Fano threefolds with 2-torus action.…”

“…By Lemma 3.4, X (S) is a toric variety. [6,Appendix] this criterion is fulfilled for all smooth complete rational Tvarieties of complexity one. Hence, they are covered by affine charts isomorphic to affine spaces.…”

“…h Q (v) = u, v + a. By concavity this implies u ∈ h , with h defined as in (6). We now consider h := h − u with h P (v) := h P (v) − u, v .…”

“…In [6] it was proved that for a variety with an open covering by affine spaces one obtains flexibility of the universal torsor. However, it is not clear whether the flexibility property extends to the total coordinate space.…”