Equivariant extensions of Ga-torsors over punctured surfaces

TL;DR

This paper investigates a special class of affine threefolds with additive group actions, providing a classification of those with proper Ga-actions and analyzing their geometric properties.

Contribution

It introduces a new class of algebraic varieties with additive group actions and provides a complete classification for a specific subclass with isolated degenerate fibers.

Findings

Classified threefolds with proper Ga-actions and isolated degenerate fibers.

Established fundamental properties of these varieties.

Constructed examples illustrating their complex geometry.

Abstract

Motivated by the study of the structure of algebraic actions the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X $\rightarrow$ X//Ga restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper Ga-actions, whose algebraic quotient morphisms $\pi$ : X $\rightarrow$ X//Ga are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A 2 when equipped with their reduced structures.