On the Stabilisation of Rational Surface Maps

Richard A. P. Birkett

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arXiv · 2110.00095

On the Stabilisation of Rational Surface Maps

Richard A. P. Birkett

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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.

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Richard A. P. Birkett. 2021-09-30. On the Stabilisation of Rational Surface Maps. https://arxiv.org/abs/2110.00095

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