Equivariant compactifications of two-dimensional algebraic groups

TL;DR

This paper classifies certain algebraic group actions on the projective plane and identifies all del Pezzo surfaces that serve as equivariant compactifications of these actions, advancing understanding in algebraic geometry and rational point distribution.

Contribution

It provides a comprehensive classification of generically transitive actions of semidirect products on the projective plane and identifies all relevant del Pezzo surfaces as equivariant compactifications.

Findings

Classification of group actions on the projective plane.

Identification of del Pezzo surfaces as equivariant compactifications.

Insights into the distribution of rational points on algebraic surfaces.

Abstract

We classify generically transitive actions of semidirect products of an additive and a multiplicative group on the projective plane. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for such semidirect products.