We look at algebraic embeddings of the complex Cremona group in variables to the group of birational transformations of an algebraic variety . First we study geometrical properties of an example of an embedding of into that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of into , where is a variety of dimension 3 and generalize this result partially to algebraic embeddings of into , where the dimension of is , for arbitrary . In particular, this yields a classification of all algebraic -actions on smooth projective varieties of dimension that can be extended to rational actions of .
On s’intéresse aux plongements algébriques du groupe de Cremona complexe à variables dans des groupes de transformations birationnelles d’une varété algébrique . D’abord on regarde un plongement de dans qui était découvert par Gizatullin. Puis on donne une classification de tous les plongements algébriques de dans pour des variétés de dimension et on généralise partiellement ce résultat aux plongements algébriques de dans , où la dimension de est (pour tout ). On obtient notamment une classification de toutes les action régulières de sur des variétés projectives lisses de dimension qui s’étendent vers des actions rationnelles de .
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Keywords: Cremona group, rational group actions, algebraic group actions
Mot clés : Groupe de Cremona, actions rationnelles des groupes
Urech, Christian 1
@article{AIF_2018__68_1_53_0, author = {Urech, Christian}, title = {On homomorphisms between {Cremona} groups}, journal = {Annales de l'Institut Fourier}, pages = {53--100}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {1}, year = {2018}, doi = {10.5802/aif.3151}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3151/} }
TY - JOUR AU - Urech, Christian TI - On homomorphisms between Cremona groups JO - Annales de l'Institut Fourier PY - 2018 SP - 53 EP - 100 VL - 68 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3151/ DO - 10.5802/aif.3151 LA - en ID - AIF_2018__68_1_53_0 ER -
%0 Journal Article %A Urech, Christian %T On homomorphisms between Cremona groups %J Annales de l'Institut Fourier %D 2018 %P 53-100 %V 68 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3151/ %R 10.5802/aif.3151 %G en %F AIF_2018__68_1_53_0
Urech, Christian. On homomorphisms between Cremona groups. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 53-100. doi : 10.5802/aif.3151. https://aif.centre-mersenne.org/articles/10.5802/aif.3151/
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