$\mathbb{A}^1_*$-fibrations on affine threefolds

TL;DR

This paper studies the structure of affine threefolds fibred by the affine line minus one point, exploring when such fibrations correspond to algebraic torus quotients and analyzing their geometric and topological properties.

Contribution

It provides a detailed analysis of affine threefolds with $\\mathbb{A}^1_*$-fibrations, including conditions for these fibrations to be torus quotients and their geometric implications.

Findings

Characterization of when fibrations are torus quotients

Topological properties of affine threefolds with such fibrations

Conditions under which the fibration structure aligns with algebraic torus actions

Abstract

The affine line minus one point is the underlying space of the algebraic torus of dimension one. However the fibration of an affine algebraic threefold by the affine line minus one point is not always the quotient morphism of the threefold by the algebraic torus. We investigate the structure of affine algebraic threefolds having fibrations by the affine line minus one point. Our results cover not only algebro-geometric but also topological properties of the threefolds. We also consider when the fibration by the affine line minus one point becomes the quotient morphism by an algebraic torus of dimension one.