On the Stabilisation of Rational Surface Maps

TL;DR

This paper investigates conditions under which rational surface maps can be conjugated to algebraically stable maps, establishing minimal conjugacies and providing examples where stability cannot be achieved solely by blow-ups.

Contribution

It introduces the concept of minimal birational conjugacy to achieve algebraic stability and demonstrates stability via graph lifting, with examples showing limitations of blow-up methods.

Findings

Existence of minimal conjugacy when achieved by a birational morphism

Stable conjugacy obtained through repeated graph lifting for birational maps

Example of conjugation to stability not achievable solely by blow-ups

Abstract

The dynamics of a rational surface map $f : X \dashrightarrow X$ are easier to analyse when $f$ is `algebraically stable'. Here we investigate when and how this condition can be achieved by conjugating $f$ with a birational change of coordinates. We show that if this can be done with a birational morphism, then there is a minimal such conjugacy. For birational $f$ we also show that repeatedly lifting $f$ to its graph gives a stable conjugacy. Finally, we give an example in which $f$ can be birationally conjugated to a stable map, but the conjugacy cannot be achieved solely by blowing up. La dynamique d'une application rationnelle $f : X \dashrightarrow X$ sur une surface est plus simple \`a analyser lorsque $f$ est `alg\'ebriquement stable'. Dans cet article nous \'etudions comment la stabilit\'e peut \^etre r\'ealis\'ee en conjuguant $f$ par un changement de variable birationnel. Nous montrons que si cela peut \^etre r\'ealis\'ee avec une morphisme birationnelle, il existe une telle conjugaison minimale. Pour $f$ birationnelle, nous montrons aussi que l'on obtient une conjugaison stable par rel\'evement successif au graphe. Nous donnons enfin un exemple dans lequel $f$ peut \^etre conjugu\'ee birationnellement \`a une application stable, mais la conjugu\'ee ne peut pas \^etre obtenue uniquement par \'eclatement.