no. 1
Relations in the Cremona group over a perfect field
[Relations dans le groupe de Cremona sur un corps parfait]
Schneider, Julia1
1 Universität Basel Departement Mathematik und Informatik Spiegelgasse 1 CH-4051 Basel (Switzerland)
Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 1-42.

Pour les corps parfaits k qui satisfont [k ¯:k]>2, nous construisons de nouveaux sous-groupes distingués du groupe de Cremona du plan et nous donnons une preuve élémentaire de sa non-simplicité en suivant la mélodie de la preuve récente de Blanc, Lamy et Zimmermann du fait que le groupe de Cremona de rang n sur les (sous-corps des) nombres complexes n’est pas simple pour n≥3.

For perfect fields k satisfying [k ¯:k]>2, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona group of rank n over (subfields of) the complex numbers is not simple for n≥3.

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Révisé le :
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DOI : 10.5802/aif.3463
Classification : 14E07, 14G27, 14E05, 14J26
Keywords: Cremona groups, normal subgroups, relations, conic bundles, Sarkisov links, Galois action, non-closed fields
Mot clés : groupes de Cremona, sous-groupes normaux, relations, fibrés en coniques, Sarkisov links, action du groupe de Galois, corps non-clos

Schneider, Julia 1

1 Universität Basel Departement Mathematik und Informatik Spiegelgasse 1 CH-4051 Basel (Switzerland)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Schneider, Julia. Relations in the Cremona group over a perfect field. Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 1-42. doi : 10.5802/aif.3463. https://aif.centre-mersenne.org/articles/10.5802/aif.3463/

[1] Asgarli, Shamil; Lai, Kuan-Wen; Nakahara, Masahiro; Zimmermann, Susanna Biregular Cremona transformations of the plane (2019) (https://arxiv.org/abs/1910.05302)

[2] Blanc, Jérémy Linearisation of finite Abelian subgroups of the Cremona group of the plane, Groups Geom. Dyn., Volume 3 (2009), pp. 215-266 | DOI | MR | Zbl

[3] Blanc, Jérémy; Lamy, Stéphane; Zimmermann, Susanna Quotients of higher-dimensional Cremona groups, Acta Math., Volume 226 (2021) no. 2, pp. 211-318 | DOI | MR | Zbl

[4] Blanc, Jérémy; Yasinsky, Egor Quotients of groups of birational transformations of cubic del Pezzo fibrations, J. Éc. Polytech., Math., Volume 7 (2020), pp. 1089-1112 | DOI | MR | Zbl

[5] Cantat, Serge; Lamy, Stéphane Normal subgroups in the Cremona group, Acta Math., Volume 210 (2013) no. 1, pp. 31-94 (With an appendix by Yves de Cornulier) | DOI | MR | Zbl

[6] Castelnuovo, Guido Le transformazioni generatrici del gruppo Cremoniano nel piano, Torino Atti, Volume 36 (1901), pp. 861-874 | Zbl

[7] Corti, Alessio Factoring birational maps of threefolds after Sarkisov. Appendix: Surfaces over nonclosed fields., J. Algebr. Geom., Volume 4 (1995) no. 2, p. 223-254, appendix 248–254 | Zbl

[8] Dolgachev, Igor V. Classical algebraic geometry. A modern view, Cambridge University Press, 2012, xii+639 pages | DOI

[9] Higgins, P. J. Presentations of groupoids, with applications to groups, Math. Proc. Camb. Philos. Soc., Volume 60 (1964) no. 1, pp. 07-20 | DOI | MR | Zbl

[10] Iskovskikh, Vasily A. Factorization of birational maps of rational surfaces from the viewpoint of Mori theory, Russ. Math. Surv., Volume 51 (1996), pp. 585-652 | DOI | Zbl

[11] Iskovskikh, Vasily A.; Kabdykairov, Farkhat K.; Tregub, Semion L. Relations in a two-dimensional Cremona group over a perfect field, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 57 (1993) no. 3, pp. 3-69 | DOI | MR | Zbl

[12] Kollár, János Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Springer, 1999

[13] Lamy, Stéphane; Zimmermann, Susanna Signature morphisms from the Cremona group over a non-closed field, J. Eur. Math. Soc., Volume 22 (2020) no. 10, pp. 3133-3173 | DOI | MR | Zbl

[14] Lang, Serge Algebra, Graduate Texts in Mathematics, Springer, 2005

[15] Lonjou, Anne Non simplicité du groupe de Cremona sur tout corps, Ann. Inst. Fourier, Volume 66 (2016) no. 5, pp. 2021-2046 | DOI | Numdam | MR | Zbl

[16] Noether, Max Ueber Flächen, welche Schaaren rationaler Curven besitzen, Math. Ann., Volume 3 (1870) no. 2, pp. 161-227 | DOI | MR | Zbl

[17] Schupp, Paul E. A survey of SQ-universality, Conference on Group Theory (Univ. Wisconsin-Parkside, Kenosha, Wis., 1972) (Lecture Notes in Mathematics), Volume 319, Springer, 1973, pp. 183-188 | DOI | MR | Zbl

[18] Shepherd-Barron, Nicholas I. Some effectivity questions for plane Cremona transformations (2013) (https://arxiv.org/abs/1311.6608)

[19] Zikas, Sokratis Rigid birational involutions of ℙ 3 and cubic threefolds (2021) (https://arxiv.org/abs/2111.04711)

[20] Zimmermann, Susanna The Abelianization of the real Cremona group, Duke Math. J., Volume 167 (2018) no. 2, pp. 211-267 | MR | Zbl

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