Normal equivariant compactifications of G^2_a with Picard number one

TL;DR

This paper classifies all normal G^2_a-surfaces with Picard number one, analyzes their singularities, and explores their G^2_a-structures, including moduli spaces, answering key questions in algebraic geometry.

Contribution

It provides a complete classification of G^2_a-surfaces with Picard number one and characterizes their singularities and G^2_a-structures, including moduli.

Findings

Classified all such surfaces with Picard number one.

Determined which surfaces have log canonical or log terminal singularities.

Identified the existence of moduli of G^2_a-structures on these surfaces.

Abstract

We classify all normal G^2_a-surfaces with Picard number one, and characterize which of these surfaces have at worst log canonical, and which have at worst log terminal singularities, answering a question of Hassett and Tschinkel (Int. Math. Res. Not., 1999). We also find all G^2_a-structures on these surfaces and show that these surfaces and their minimal desingularizations have the same G^2_a-structures (modulo equivalence of G^2_a-actions). In particular, we show that some of these surfaces admit one dimensional moduli of G^2_a-structures, answering another question of Hassett and Tschinkel (Int. Math. Res. Not., 1999).