Unipotent Group Actions on Del Pezzo Cones

TL;DR

This paper investigates the presence of unipotent group actions on affine cones over del Pezzo surfaces, showing that such actions exist for degree ≥4 but not for degree ≤2.

Contribution

It establishes a clear dichotomy in unipotent group actions on affine cones over del Pezzo surfaces based on their degree, extending previous results on automorphism groups.

Findings

Unipotent group actions exist for degree ≥4 del Pezzo surfaces.

No non-trivial unipotent actions for degree ≤2 del Pezzo surfaces.

Automorphism groups are infinite dimensional for degree ≥4 cases.

Abstract

In a previous paper we established that for any del Pezzo surface Y of degree at least 4, the affine cone X over Y embedded via a pluri-anticanonical linear system admits an effective Ga-action. In particular, the group Aut(X) is infinite dimensional. In contrast, we show in this note that for a del Pezzo surface Y of degree at most 2 the generalized cones X as above do not admit any non-trivial action of a unipotent algebraic group.