On the Danilov-Gizatullin Isomorphism Theorem

Abstract

A Danilov–Gizatullin surface is a normal affine surface $V={\Sigma }_d\setminus S$, which is a complement to an ample section $S$ in a Hirzebruch surface ${\Sigma }_d$. By a surprising result due to Danilov and Gizatullin [DaGi], $V$ depends only on $n=S^2$ and neither on $d$ nor on $S$. In this note we provide a new and simple proof of this Isomorphism Theorem.