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.
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.
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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
@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 TI - Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 233 EP - 252 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.217/ DO - 10.5802/jep.217 LA - en ID - JEP_2023__10__233_0 ER -
%0 Journal Article %A Sokratis Zikas %T Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 233-252 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.217/ %R 10.5802/jep.217 %G en %F JEP_2023__10__233_0
Sokratis Zikas. Rigid birational involutions of $\mathbb{P}^3$ and cubic threefolds. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 233-252. doi : 10.5802/jep.217. https://jep.centre-mersenne.org/articles/10.5802/jep.217/
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