On the Danilov-Gizatullin Isomorphism Theorem
  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. 
Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg