On the Danilov-Gizatullin Isomorphism Theorem

TL;DR

This paper presents a new, simplified proof of the Danilov-Gizatullin Isomorphism Theorem, which states that certain affine surfaces depend only on the self-intersection number of a section, not on other parameters.

Contribution

It offers a novel, straightforward proof of the theorem, clarifying the dependence of the surface on the self-intersection number alone.

Findings

The proof confirms the surface's dependence solely on the self-intersection number.

Simplifies understanding of the Danilov-Gizatullin Isomorphism Theorem.

Enhances theoretical foundations of affine surface classification.

Abstract

A Danilov-Gizatullin surface is a normal affine surface V, which is a complement to an ample section S in a Hirzebruch surface of index d. By a surprising result due to Danilov and Gizatullin, V depends only on the self-intersection number of S and neither on d nor on S. In this note we provide a new and simple proof of this Isomorphism Theorem.