Factorial threefolds with Ga-actions

TL;DR

This paper investigates factorial threefolds with Ga-actions in the context of the affine cancellation problem, exploring when such varieties are isomorphic or serve as counterexamples, with a focus on locally trivial Ga-actions and Ga-bundles.

Contribution

It introduces a new class of factorial threefolds with Ga-actions, analyzes their properties, and discusses their potential as counterexamples to the affine cancellation problem.

Findings

Some threefolds with Ga-actions are isomorphic as abstract varieties.

Counterexamples to the cancellation problem remain unknown within this class.

Locally trivial Ga-actions are key to understanding these varieties.

Abstract

The affine cancellation problem, which asks whether complex affine varieties with isomorphic cylinders are themselves isomorphic, has a positive solution for two dimensional varieties whose coordinate rings are unique factorization domains, in particular for the affine plane, but counterexamples are found within normal surfaces Danielewski surfaces and factorial threefolds of logarithmic Kodaira dimension equal to 1. The latter are therefore remote from the affine three-space, the first unknown case where the base of one cylinder is an affine space. Locally trivial Ga-actions play a significant role in these examples. Threefolds admitting free Ga-actions are discussed, especially a class of varieties with negative logarithmic Kodaira dimension which are total spaces of nonisomorphic Ga-bundles. Some members of the class are shown to be isomorphic as abstract varieties, but it is unknown whether any members of the class constitute counterexamples to cancellation.