Group actions on affine cones

TL;DR

This paper investigates which affine cones over smooth projective varieties admit non-standard algebraic group actions, providing specific results for del Pezzo surfaces and a general geometric criterion for broader cases.

Contribution

It identifies affine cones over certain del Pezzo surfaces that admit non-standard group actions and offers a geometric criterion applicable to other varieties.

Findings

Affine cones over del Pezzo surfaces of degree ≥ 4 admit such actions.

A geometric criterion for the existence of non-standard group actions.

The question for cubic surfaces remains open with current methods.

Abstract

We address the following question: Determine the affine cones over smooth projective varieties which admit an action of a connected algebraic group different from the standard C*-action by scalar matrices and its inverse action. We show in particular that the affine cones over anticanonically embedded smooth del Pezzo surfaces of degree at least 4 possess such an action. A question by Flenner and the third author whether this is also true for cubic surfaces, occurs to be out of reach for our methods. Nevertheless, we provide a general geometric criterion that could be helpful also in this case.