Rational surfaces with a large group of automorphisms

TL;DR

This paper classifies rational surfaces with maximal automorphism groups, answering a question from 1928, and explores related classifications of non-rational surfaces with large automorphism groups.

Contribution

It provides a complete classification of rational surfaces with the largest automorphism groups and discusses the classification of non-rational surfaces with large automorphism groups.

Findings

Classification of rational surfaces with maximal automorphism groups

Connection to periodic orbits of Coxeter group actions

Outline of non-rational surfaces with large automorphism groups

Abstract

We classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in terms of periodic orbits of a birational action of an infinite Coxeter group on ordered point sets in the projective plane modulo projective equivalence. We also outline the classification of non-rational surfaces with large automorphism groups.