Quotients of groups of birational transformations of cubic del Pezzo fibrations
[Quotients de groupes de transformations birationnelles de fibrations en del Pezzo cubiques]
Journal de l’École polytechnique — Mathématiques, Tome 7 (2020) , pp. 1089-1112.

Nous démontrons que le groupe des transformations birationnelles d’une fibration de del Pezzo de degré 3 sur une courbe n’est pas simple, en donnant un homomorphisme de groupes surjectif vers un produit libre d’une infinité de groupes d’ordre 2. Par conséquent, nous obtenons que le groupe de Cremona de rang 3 n’est pas engendré par les applications birationnelles qui préservent une fibration rationnelles. De plus, le sous-groupe de Bir( 3 ) engendré par tous les sous-groupes algébriques connexes est un sous-groupe distingué propre.

We prove that the group of birational transformations of a del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, the subgroup of Bir( 3 ) generated by all connected algebraic subgroups is a proper normal subgroup.

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DOI : https://doi.org/10.5802/jep.136
Classification : 14E07,  14E05,  14E30,  14J45,  14M22
Mots clés : Fibrations de del Pezzo, groupe de Cremona, homomorphismes de groupes, quotients de groupes, transformations birationnelles, genre
@article{JEP_2020__7__1089_0,
     author = {J\'er\'emy Blanc and Egor Yasinsky},
     title = {Quotients of groups of birational transformations of cubic del {Pezzo} fibrations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1089--1112},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.136},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.136/}
}
Jérémy Blanc; Egor Yasinsky. Quotients of groups of birational transformations of cubic del Pezzo fibrations. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020) , pp. 1089-1112. doi : 10.5802/jep.136. https://jep.centre-mersenne.org/articles/10.5802/jep.136/

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