Automorphism groups and anti-pluricanonical curves

TL;DR

This paper proves the existence of anti-pluricanonical curves on certain rational surfaces with infinite automorphism groups and minimal pairs, confirming longstanding conjectures and strengthening the Tits alternative theorem.

Contribution

It establishes the existence of anti-pluricanonical curves on minimal rational surfaces with infinite automorphism groups, confirming conjectures by McMullen and others.

Findings

Existence of anti-pluricanonical curves on minimal rational surfaces with infinite automorphism groups.

Validation of a conjecture by McMullen (2005).

A strengthened form of the Tits alternative theorem.

Abstract

We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z . Z (semi-direct product), provided that the pair (X, G) is minimal. This was conjectured by Curtis T. McMullen (2005) and further traced back to Marat Gizatullin and Brian Harbourne (1987). We also prove (perhaps) the strongest form of the famous Tits alternative theorem.