Algebraic density property of Danilov-Gizatullin surfaces

TL;DR

This paper studies Danilov-Gizatullin surfaces, showing they can be represented as quotients of affine threefolds and proving their algebraic vector fields are generated by complete ones, revealing their algebraic density property.

Contribution

It demonstrates that Danilov-Gizatullin surfaces have the algebraic density property by expressing them as quotients and analyzing their vector fields.

Findings

V is a quotient of an affine threefold by a torus action.

The Lie algebra generated by complete vector fields equals all algebraic vector fields on V.

V's isomorphism class depends only on the self-intersection number S^2.

Abstract

A Danilov-Gizatullin surface is an affine surface $V$ which is the complement of an ample section $S$ of a Hirzebruch surface. The remarkable theorem of Danilov and Gizatullin states that the isomorphism class of $V$ depends only on the self-intersection number $S^2$. In this paper we apply their theorem to present $V$ as the quotient of an affine threefold by a torus action, and to prove that the Lie algebra generated by the complete algebraic vector fields on $V$ coincides with the set of all algebraic vector fields.