Singular del Pezzo surfaces that are equivariant compactifications

TL;DR

This paper classifies singular del Pezzo surfaces that can be compactified with algebraic group actions, aiding in proving Manin's conjecture, and provides an example involving a semidirect product of algebraic groups.

Contribution

It identifies which singular del Pezzo surfaces are equivariant compactifications of G_a^2 and presents a novel example involving a semidirect product of G_a and G_m.

Findings

Classification of singular del Pezzo surfaces as equivariant compactifications

Identification of a specific quartic del Pezzo surface with a semidirect product structure

Support for Manin's conjecture through these classifications

Abstract

We determine which singular del Pezzo surfaces are equivariant compactifications of G_a^2, to assist with proofs of Manin's conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of a semidirect product of G_a and G_m.