Del Pezzo quintics as equivariant compactifications of vector groups

TL;DR

This paper classifies smooth and mildly singular quintic del Pezzo varieties that admit dense orbits of abelian unipotent groups, revealing uniqueness of such actions and their geometric implications.

Contribution

It provides a complete classification of abelian unipotent group actions with dense orbits on quintic del Pezzo varieties, including smooth and mildly singular cases, and establishes uniqueness up to automorphisms.

Findings

Characterization of smooth forms admitting abelian unipotent actions

Uniqueness of the group action up to automorphisms

Classification of mildly singular quintic del Pezzo threefolds and surfaces

Abstract

We study faithful actions with a dense orbit of abelian unipotent groups on quintic del Pezzo varieties over a field of characteristic zero. Such varieties are forms of linear sections of the Grassmannian of planes in a 5-dimensional vector space. We characterize which smooth forms admit these types of actions and show that in case of existence, the action is unique up to equivalence by automorphisms. We also give a similar classification for mildly singular quintic del Pezzo threefolds and surfaces.