no. 2
Automorphisms of real del Pezzo surfaces and the real plane Cremona group
[Automorphismes de surfaces de del Pezzo réelles et le groupe de Cremona du plan projectif réel]
Yasinsky, Egor1
1 Universität Basel Departement Mathematik und Informatik Spiegelgasse 1 4051 Basel, Switzerland
Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 831-899.

On étudie les groupes d’automorphismes des surfaces de del Pezzo réelles, en se concentrant sur les groupes finis qui agissent avec un nombre invariant de Picard égal à 1. En conséquence, on obtient une bonne part de la classification des sous-groupes finis du groupe de Cremona du plan réel.

We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting with invariant Picard number equal to one. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3460
Classification : 14E05, 14E07, 14P05, 14J26, 14J50, 20D99
Keywords: Cremona group, conic bundle, del Pezzo surface, automorphism group, real algebraic surface.
Mot clés : Groupe de Cremona, fibré conique, surface de del Pezzo, surface algébrique réelle.
Yasinsky, Egor 1

1 Universität Basel Departement Mathematik und Informatik Spiegelgasse 1 4051 Basel, Switzerland
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2022__72_2_831_0,
     author = {Yasinsky, Egor},
     title = {Automorphisms of real del {Pezzo} surfaces and the real plane {Cremona} group},
     journal = {Annales de l'Institut Fourier},
     pages = {831--899},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {2},
     year = {2022},
     doi = {10.5802/aif.3460},
     zbl = {07554670},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3460/}
}
TY  - JOUR
AU  - Yasinsky, Egor
TI  - Automorphisms of real del Pezzo surfaces and the real plane Cremona group
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 831
EP  - 899
VL  - 72
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3460/
DO  - 10.5802/aif.3460
LA  - en
ID  - AIF_2022__72_2_831_0
ER  - 
%0 Journal Article
%A Yasinsky, Egor
%T Automorphisms of real del Pezzo surfaces and the real plane Cremona group
%J Annales de l'Institut Fourier
%D 2022
%P 831-899
%V 72
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3460/
%R 10.5802/aif.3460
%G en
%F AIF_2022__72_2_831_0
Yasinsky, Egor. Automorphisms of real del Pezzo surfaces and the real plane Cremona group. Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 831-899. doi : 10.5802/aif.3460. https://aif.centre-mersenne.org/articles/10.5802/aif.3460/

[1] Beauville, Arnaud p-elementary subgroups of the Cremona group, J. Algebra, Volume 314 (2007) no. 2, pp. 553-564 | DOI | MR | Zbl

[2] Beauville, Arnaud Finite subgroups of PGL 2 (K), Vector bundles and complex geometry (Contemp. Math.), Volume 522, Amer. Math. Soc., Providence, RI, 2010, pp. 23-29 | DOI | MR | Zbl

[3] Birkar, Caucher Singularities of linear systems and boundedness of Fano varieties (2016) (https://arxiv.org/pdf/)

[4] Blanc, Jérémy Linearisation of finite abelian subgroups of the Cremona group of the plane, Groups Geom. Dyn., Volume 3 (2009) no. 2, pp. 215-266 | DOI | MR | Zbl

[5] Blichfeldt, H. F. Finite collineation groups, The Univ. Chicago Press, 1917

[6] Bredon, Glen E. Introduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press, New York-London, 1972, xiii+459 pages | MR

[7] Bujalance, Emilio; Etayo, José Javier; Gamboa, José Manuel Automorphism groups of real algebraic curves of genus 3, Proc. Japan Acad. Ser. A Math. Sci., Volume 62 (1986) no. 1, pp. 40-42 | DOI | MR | Zbl

[8] Carter, Roger W. Conjugacy classes in the Weyl group, Compositio Math., Volume 25 (1972), pp. 1-59 | Numdam | MR | Zbl

[9] Chelʼtsov, Ivan A. Two local inequalities, Izv. Ross. Akad. Nauk Ser. Mat., Volume 78 (2014) no. 2, pp. 167-224 | DOI | MR

[10] Conway, John H.; Smith, Derek A. On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd., Natick, MA, 2003, xii+159 pages | DOI | MR | Zbl

[11] Dolgachev, Igor V. Classical algebraic geometry, Cambridge University Press, Cambridge, 2012, xii+639 pages (A modern view) | DOI | MR | Zbl

[12] Dolgachev, Igor V.; Duncan, Alexander Automorphisms of cubic surfaces in positive characteristic, Izv. Ross. Akad. Nauk Ser. Mat., Volume 83 (2019) no. 3, pp. 15-92 | DOI | MR | Zbl

[13] Dolgachev, Igor V.; Iskovskikh, Vasily A. Finite subgroups of the plane Cremona group, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I (Progr. Math.), Volume 269, Birkhäuser Boston, Boston, MA, 2009, pp. 443-548 | DOI | MR | Zbl

[14] Dolgachev, Igor V.; Iskovskikh, Vasily A. On elements of prime order in the plane Cremona group over a perfect field, Int. Math. Res. Not. IMRN (2009) no. 18, pp. 3467-3485 | DOI | MR | Zbl

[15] Hosoh, Toshio Automorphism groups of cubic surfaces, J. Algebra, Volume 192 (1997) no. 2, pp. 651-677 | DOI | MR | Zbl

[16] Iskovskikh, Vasilii A. Rational surfaces with a pencil of rational curves, Mat. Sb. (N.S.), Volume 74 (116) (1967), pp. 608-638 | MR

[17] Iskovskikh, Vasilii A. Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Uspekhi Mat. Nauk, Volume 51 (1996) no. 4(310), pp. 3-72 | DOI | MR | Zbl

[18] James, Gordon; Liebeck, Martin Representations and characters of groups, Cambridge University Press, New York, 2001, viii+458 pages | DOI | MR | Zbl

[19] Kollar, Janos Real Algebraic Surfaces (1997) (Notes of the 1997 Trento summer school lectures)

[20] Manin, Yu I. Cubic forms. Algebra, geometry, arithmetic, North-Holland Mathematical Library, 4, North-Holland Publishing Co., Amsterdam, 1986, x+326 pages (Translated from the Russian by M. Hazewinkel) | MR | Zbl

[21] Patera, Jiri; Saint-Aubin, Yvan; Zassenhaus, Hans Finite subgroups of the generalized Lorentz groups O(p,q), J. Math. Phys., Volume 21 (1980) no. 2, pp. 234-239 | DOI | MR | Zbl

[22] Polyakova, Yu M. Factoring birational maps of rational surfaces over the field of real numbers, Fundam. Prikl. Mat., Volume 3 (1997) no. 2, pp. 519-547 | MR | Zbl

[23] Popov, Vladimir L. On the Makar–Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, Affine algebraic geometry (CRM Proc. Lecture Notes), Volume 54, Amer. Math. Soc., Providence, RI, 2011, pp. 289-311 | DOI | MR | Zbl

[24] Prokhorov, Yu G. Simple finite subgroups of the Cremona group of rank 3, J. Algebraic Geom., Volume 21 (2012) no. 3, pp. 563-600 | DOI | MR | Zbl

[25] Prokhorov, Yu G. On G-Fano threefolds, Izv. Ross. Akad. Nauk Ser. Mat., Volume 79 (2015) no. 4, pp. 159-174 | DOI | MR | Zbl

[26] Prokhorov, Yu G. Quasi-simple finite groups of essential dimension 3, Fundam. Prikl. Mat., Volume 22 (2019) no. 4, pp. 189-199 | DOI | Zbl

[27] Prokhorov, Yu G.; Shramov, Constantin Jordan property for Cremona groups, Amer. J. Math., Volume 138 (2016) no. 2, pp. 403-418 | DOI | MR | Zbl

[28] Prokhorov, Yu G.; Shramov, Constantin Jordan constant for Cremona group of rank 3, Mosc. Math. J., Volume 17 (2017) no. 3, pp. 457-509 | DOI | MR | Zbl

[29] Robayo, Maria Fernanda Prime order birational diffeomorphisms of the sphere, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 16 (2016) no. 3, pp. 909-970 | DOI | MR | Zbl

[30] Robayo, Maria Fernanda; Zimmermann, Susanna Infinite algebraic subgroups of the real Cremona group, Osaka J. Math., Volume 55 (2018) no. 4, pp. 681-712 | MR | Zbl

[31] Segre, Beniamino The Non-singular Cubic Surfaces, Oxford University Press, Oxford, 1942, xi+180 pages | MR | Zbl

[32] Trepalin, Andrey Quotients of conic bundles, Transform. Groups, Volume 21 (2016) no. 1, pp. 275-295 | DOI | MR | Zbl

[33] Trepalin, Andrey Quotients of del Pezzo surfaces of degree 2, Mosc. Math. J., Volume 18 (2018) no. 3, pp. 557-597 | DOI | MR | Zbl

[34] Trepalin, Andrey Quotients of del Pezzo surfaces, Internat. J. Math., Volume 30 (2019) no. 12, p. 1950068, 40 | DOI | MR | Zbl

[35] Tsygankov, Vladimir I. The conjugacy classes of finite nonsolvable subgroups in the plane Cremona group, Adv. Geom., Volume 13 (2013) no. 2, pp. 323-347 | DOI | MR | Zbl

[36] Voskresenskiĭ, Valentin E. Algebraic groups and their birational invariants, Translations of Mathematical Monographs, 179, American Mathematical Society, Providence, RI, 1998, xiv+218 pages (Translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ]) | DOI | MR | Zbl

[37] Wall, Charles T. C. Stability, pencils and polytopes, Bull. London Math. Soc., Volume 12 (1980) no. 6, pp. 401-421 | DOI | MR | Zbl

[38] Wall, Charles T. C. Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math., Volume 375/376 (1987), pp. 47-66 | DOI | MR | Zbl

[39] Yasinsky, Egor Subgroups of odd order in the real plane Cremona group, J. Algebra, Volume 461 (2016), pp. 87-120 | DOI | MR | Zbl

[40] Yasinsky, Egor The Jordan constant for Cremona group of rank 2, Bull. Korean Math. Soc., Volume 54 (2017) no. 5, pp. 1859-1871 | DOI | MR | Zbl

[41] Zimmermann, Bruno P. On finite groups acting on spheres and finite subgroups of orthogonal groups, Sib. Èlektron. Mat. Izv., Volume 9 (2012), pp. 1-12 | MR | Zbl

Cité par Sources :