Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)

TL;DR

This paper explains Gizatullin's classification of rational surfaces with parabolic automorphisms, detailing the original proof and recent developments, making complex concepts accessible to non-experts.

Contribution

It provides an accessible exposition of Gizatullin's classification and introduces recent advancements by Cantat and Dolgachev in the study of parabolic automorphisms.

Findings

Rational surfaces with parabolic automorphisms admit invariant elliptic fibrations.

Gizatullin's original proof is thoroughly explained for non-experts.

Recent work extends understanding of automorphism actions on these surfaces.

Abstract

In 1980, Gizatullin classified rational surfaces endowed with an automorphism whose action on the Neron-Severi group is parabolic: these surfaces are endowed with an elliptic fibration invariant by the automorphism. The aim of this expository paper is to present for non-experts the details of Gizatullin's original proof, and to provide an introduction to a recent paper by Cantat and Dolgachev.