Additive actions on projective surfaces with a finite number of orbits

TL;DR

This paper classifies certain projective surfaces with singularities that admit additive group actions with finitely many orbits, revealing new examples of such actions and answering an open question in the field.

Contribution

It provides a classification of projective surfaces with du Val singularities admitting additive actions with finitely many orbits and constructs examples with continuous families of non-isomorphic actions.

Findings

Classified projective surfaces with du Val singularities admitting additive actions.

Constructed examples with 1-parameter families of non-isomorphic additive actions.

Answered an open question by Hassett and Tschinkel.

Abstract

An additive action on an algebraic variety is an effective action of the vector group with an open orbit. We describe projective surfaces with du Val singularities that admit an additive action with a finite number of orbits. In particular, we provide examples of projective surfaces with 1-parameter families of pairwise non-isomorphic additive actions, which answers the question by Hassett and Tschinkel.