[Involutions birationnelles rigides de et de cubiques lisses de dimension ]
Nous construisons des familles d’involutions birationnelles sur ou sur une cubique lisse de dimension qui ne s’intègrent pas dans une relation élémentaire non triviale de liens de Sarkisov. En conséquence, nous construisons de nouveaux homomorphismes à partir de leur groupe de transformations birationnelles, redémontrant de manière effective leur non-simplicité. Nous prouvons également que ces groupes admettent une structure de produit libre. Enfin, nous produisons des automorphismes de ces groupes qui ne sont pas engendrés par des automorphismes intérieurs et des automorphismes de corps.
We construct families of birational involutions on or on a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.
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Keywords: Cremona groups, Sarkisov links, rank $3$ fibrations, elementary relations, cubic 3-folds
Mot clés : Groupes de Cremona, liens de Sarkisov, fibrations de rang $3$, relations élémentaires, cubiques lisses de dimension $3$
Sokratis Zikas 1
CC-BY 4.0
@article{JEP_2023__10__233_0,
author = {Sokratis Zikas},
title = {Rigid birational involutions of $\mathbb{P}^3$ and cubic~threefolds},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {233--252},
publisher = {\'Ecole polytechnique},
volume = {10},
year = {2023},
doi = {10.5802/jep.217},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.217/}
}
TY - JOUR
AU - Sokratis Zikas
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JO - Journal de l’École polytechnique — Mathématiques
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EP - 252
VL - 10
PB - École polytechnique
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Sokratis Zikas. Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 233-252. doi : 10.5802/jep.217. https://jep.centre-mersenne.org/articles/10.5802/jep.217/
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