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 à analyser lorsque $f$ est « algébriquement stable ». Dans cet article nous étudions comment la stabilité peut être réalisée en conjuguant $f$ par un changement de variable birationnel. Nous montrons que si cela peut être réalisé avec un morphisme birationnel, il existe alors une telle conjugaison minimale. Pour $f$ birationnelle, nous montrons aussi que l’on obtient une conjugaison stable par relèvement successif au graphe. Nous donnons enfin un exemple dans lequel $f$ peut être conjuguée birationnellement à une application stable, mais la conjuguée ne peut pas être obtenue uniquement par éclatement.
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Richard A. P. Birkett 1
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@article{AFST_2025_6_34_1_47_0,
author = {Richard A. P. Birkett},
title = {On the {Stabilisation} of {Rational} {Surface} {Maps}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {47--74},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 34},
number = {1},
year = {2025},
doi = {10.5802/afst.1805},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1805/}
}
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%0 Journal Article %A Richard A. P. Birkett %T On the Stabilisation of Rational Surface Maps %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 47-74 %V 34 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1805/ %R 10.5802/afst.1805 %G en %F AFST_2025_6_34_1_47_0
Richard A. P. Birkett. On the Stabilisation of Rational Surface Maps. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 1, pp. 47-74. doi : 10.5802/afst.1805. https://afst.centre-mersenne.org/articles/10.5802/afst.1805/
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