Quotients of groups of birational transformations of cubic del Pezzo fibrations

Jérémy Blanc; Egor Yasinsky

doi:10.5802/jep.136

Quotients of groups of birational transformations of cubic del Pezzo fibrations
[Quotients de groupes de transformations birationnelles de fibrations en del Pezzo cubiques]

Jérémy Blanc ¹ ; Egor Yasinsky ¹

¹ Universität Basel, Departement Mathematik und Informatik Spiegelgasse 1, CH-4051 Basel, Switzerland

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1089-1112.

Résumé
Abstract

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 $Bir (ℙ^{3})$ 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 $Bir (ℙ^{3})$ generated by all connected algebraic subgroups is a proper normal subgroup.

Reçu le : 2019-08-05
Accepté le : 2020-08-03
Publié le : 2020-08-18

DOI : 10.5802/jep.136

Classification : 14E07, 14E05, 14E30, 14J45, 14M22
Keywords: Del Pezzo fibrations, Cremona group, group homomorphisms, group quotients, birational transformations, genus
Mot clés : Fibrations de del Pezzo, groupe de Cremona, homomorphismes de groupes, quotients de groupes, transformations birationnelles, genre

Affiliations des auteurs :

Jérémy Blanc ¹ ; Egor Yasinsky ¹

¹ Universität Basel, Departement Mathematik und Informatik Spiegelgasse 1, CH-4051 Basel, Switzerland

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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/}
}

TY  - JOUR
AU  - Jérémy Blanc
AU  - Egor Yasinsky
TI  - Quotients of groups of birational transformations of cubic del Pezzo fibrations
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2020
SP  - 1089
EP  - 1112
VL  - 7
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.136/
DO  - 10.5802/jep.136
LA  - en
ID  - JEP_2020__7__1089_0
ER  -

%0 Journal Article
%A Jérémy Blanc
%A Egor Yasinsky
%T Quotients of groups of birational transformations of cubic del Pezzo fibrations
%J Journal de l’École polytechnique — Mathématiques
%D 2020
%P 1089-1112
%V 7
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.136/
%R 10.5802/jep.136
%G en
%F JEP_2020__7__1089_0

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/

Bibliographie
Cité par

[BB73] A. Białynicki-Birula - “On fixed point schemes of actions of multiplicative and additive groups”, Topology 12 (1973), p. 99-103 | DOI | MR | Zbl

[BCDP19] J. Blanc, I. Cheltsov, A. Duncan & Y. Prokhorov - “Birational transformations of threefolds of (un)-bounded genus or gonality”, 2019 | arXiv

[BCHM10] C. Birkar, P. Cascini, C. D. Hacon & J. McKernan - “Existence of minimal models for varieties of log general type”, J. Amer. Math. Soc. 23 (2010) no. 2, p. 405-468 | DOI | MR | Zbl

[BF13] J. Blanc & J.-P. Furter - “Topologies and structures of the Cremona groups”, Ann. of Math. (2) 178 (2013) no. 3, p. 1173-1198 | DOI | MR | Zbl

[BFT19] J. Blanc, A. Fanelli & R. Terpereau - “Connected algebraic groups acting on $3$ -dimensional Mori fibrations”, 2019 | arXiv

[Bir16] C. Birkar - “Singularities of linear systems and boundedness of Fano varieties”, 2016 | arXiv | Zbl

[Bir19] C. Birkar - “Anti-pluricanonical systems on Fano varieties”, Ann. of Math. (2) 190 (2019) no. 2, p. 345-463 | DOI | MR | Zbl

[BLZ19] J. Blanc, S. Lamy & S. Zimmermann - “Quotients of higher dimensional Cremona groups”, 2019, Acta Math. (to appear) | arXiv

[Bro06] R. Brown - Topology and groupoids, BookSurge, LLC, Charleston, SC, 2006 | Zbl

[Can13] S. Cantat - “The Cremona group in two variables”, in European Congress of Mathematics, European Mathematical Society, Zürich, 2013, p. 211-225 | MR | Zbl

[Cas01] G. Castelnuovo - “Le transformazioni generatrici del gruppo Cremoniano nel piano”, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 36 (1901), p. 861-874 | Zbl

[CL13] S. Cantat & S. Lamy - “Normal subgroups in the Cremona group”, Acta Math. 210 (2013) no. 1, p. 31-94, With an appendix by Yves de Cornulier | DOI | MR | Zbl

[DI09] I. V. Dolgachev & V. A. Iskovskikh - “Finite subgroups of the plane Cremona group”, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progress in Math., vol. 269, Birkhäuser Boston, Boston, MA, 2009, p. 443-548 | DOI | MR | Zbl

[Dés19] J. Déserti - “The Cremona group and its subgroups”, 2019 | arXiv | Zbl

[Giz82] M. K. Gizatullin - “Defining relations for the Cremona group of the plane”, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982) no. 5, p. 909-970, English translation in Math. USSR 21 (1983), no. 2, p. 211–268 | MR | Zbl

[Har81] J. Harris - “A bound on the geometric genus of projective varieties”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981) no. 1, p. 35-68 | Numdam | MR | Zbl

[HK00] Y. Hu & S. Keel - “Mori dream spaces and GIT”, Michigan Math. J. 48 (2000), p. 331-348 | DOI | MR | Zbl

[HM13] C. D. Hacon & J. McKernan - “The Sarkisov program”, J. Algebraic Geom. 22 (2013) no. 2, p. 389-405 | DOI | MR | Zbl

[IKT93] V. A. Iskovskikh, F. K. Kabdykairov & S. L. Tregub - “Relations in a two-dimensional Cremona group over a perfect field”, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993) no. 3, p. 3-69, English transl. in Russian Acad. Sci. Izv. Math. 42 (1994), no. 3, p. 427–478 | DOI | Zbl

[Isk96] V. A. Iskovskikh - “Factorization of birational mappings of rational surfaces from the point of view of Mori theory”, Uspehi Mat. Nauk 51 (1996) no. 4(310), p. 3-72, English transl. in Russian Math. Surveys 51 (1996), no. 4, p. 585–652 | DOI | MR | Zbl

[Kal13] A.-S. Kaloghiros - “Relations in the Sarkisov program”, Compositio Math. 149 (2013) no. 10, p. 1685-1709 | DOI | MR | Zbl

[Kaw92] Y. Kawamata - “Boundedness of $ℚ$ -Fano threefolds”, in Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), Contemp. Math., vol. 131, American Mathematical Society, Providence, RI, 1992, p. 439-445 | MR | Zbl

[KKL16] A.-S. Kaloghiros, A. Küronya & V. Lazić - “Finite generation and geography of models”, in Minimal models and extremal rays (Kyoto, 2011), Adv. Stud. Pure Math., vol. 70, Math. Soc. Japan, Tokyo, 2016, p. 215-245 | DOI | MR | Zbl

[KM98] J. Kollár & S. Mori - Birational geometry of algebraic varieties, Cambridge Tracts in Math., vol. 134, Cambridge University Press, Cambridge, 1998 | DOI | MR | Zbl

[KMMT00] J. Kollár, Y. Miyaoka, S. Mori & H. Takagi - “Boundedness of canonical $ℚ$ -Fano 3-folds”, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000) no. 5, p. 73-77 | DOI | MR | Zbl

[Kol96] J. Kollár - Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), vol. 32, Springer-Verlag, Berlin, 1996 | DOI | MR | Zbl

[Kra18] H. Kraft - “Regularization of rational group actions”, 2018 | arXiv | Zbl

[Lon16] A. Lonjou - “Non simplicité du groupe de Cremona sur tout corps”, Ann. Inst. Fourier (Grenoble) 66 (2016) no. 5, p. 2021-2046 | DOI | Numdam | MR | Zbl

[LZ17] S. Lamy & S. Zimmermann - “Signature morphisms from the Cremona group over a non-closed field”, 2017, J. Eur. Math. Soc. (JEMS) (to appear) | arXiv

[Mat58] T. Matsusaka - “Polarized varieties, fields of moduli and generalized Kummer varieties of polarized abelian varieties”, Amer. J. Math. 80 (1958), p. 45-82 | DOI | MR | Zbl

[Mat63] H. Matsumura - “On algebraic groups of birational transformations”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 34 (1963), p. 151-155 | MR | Zbl

[Sch19] J. Schneider - “Relations in the Cremona group over perfect fields”, 2019 | arXiv

[Sob02] I. V. Sobolev - “Birational automorphisms of a class of varieties fibered by cubic surfaces”, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002) no. 1, p. 203-224 | DOI | MR | Zbl

[UZ19] C. Urech & S. Zimmermann - “A new presentation of the plane Cremona group”, Proc. Amer. Math. Soc. 147 (2019) no. 7, p. 2741-2755 | DOI | MR | Zbl

[Wei55] A. Weil - “On algebraic groups of transformations”, Amer. J. Math. 77 (1955), p. 355-391 | DOI | MR | Zbl

[Zim18] S. Zimmermann - “The Abelianization of the real Cremona group”, Duke Math. J. 167 (2018) no. 2, p. 211-267 | DOI | MR | Zbl

Cité par Sources :