Automorphisms of surfaces over fields of positive characteristic

Chen, Yifei¹ ; Shramov, Constantin²

¹ Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
² Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia, Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow, Russia

Geometry & topology, Tome 28 (2024) no. 6, pp. 2747-2791.

Résumé

We study automorphism and birational automorphism groups of varieties over fields of positive characteristic from the point of view of Jordan and $p$–Jordan property. In particular, we show that the Cremona group of rank $2$ over a field of characteristic $p>0$ is $p$–Jordan, and the birational automorphism group of an arbitrary geometrically irreducible algebraic surface is nilpotently $p$–Jordan of class at most $2$. Also, we show that the automorphism group of a smooth geometrically irreducible projective variety of nonnegative Kodaira dimension is Jordan in the usual sense.

Détail

DOI : 10.2140/gt.2024.28.2747

Keywords: birational automorphism group, $p$–Jordan property

Bibliographie

[1] L Bădescu, Algebraic surfaces, Springer (2001) | DOI

[2] T Bandman, Y G Zarhin, Jordan groups and algebraic surfaces, Transform. Groups 20 (2015) 327 | DOI

[3] L Bieberbach, Über einen Satz des Hrn. C Jordan in der Theorie der endlichen Gruppen linearer Substitutionen, Berl. Ber. 1911 (1911) 231

[4] C Birkar, Existence of flips and minimal models for 3–folds in char p, Ann. Sci. École Norm. Sup. 49 (2016) 169 | DOI

[5] C Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. 193 (2021) 347 | DOI

[6] C Birkar, J Waldron, Existence of Mori fibre spaces for 3–folds in char p, Adv. Math. 313 (2017) 62 | DOI

[7] J Blanc, J P Furter, Topologies and structures of the Cremona groups, Ann. of Math. 178 (2013) 1173 | DOI

[8] E Bombieri, D Mumford, Enriques’ classification of surfaces in char p, III, Invent. Math. 35 (1976) 197 | DOI

[9] E Bombieri, D Mumford, Enriques’ classification of surfaces in char p, II, from: "Complex analysis and algebraic geometry" (editors W L Baily Jr., T Shioda), Iwanami Shoten (1977) 23 | DOI

[10] A Borel, Linear algebraic groups, 126, Springer (1991) | DOI

[11] R Brauer, W Feit, An analogue of Jordan’s theorem in characteristic p, Ann. of Math. 84 (1966) 119 | DOI

[12] E Breuillard, An exposition of Jordan’s original proof of his theorem on finite subgroups of GLn(C), Model Theory 2 (2023) 429 | DOI

[13] E Breuillard, B Green, Approximate groups, III : The unitary case, Turkish J. Math. 36 (2012) 199 | DOI

[14] M Brion, Some structure theorems for algebraic groups, from: "Algebraic groups: structure and actions", Proc. Sympos. Pure Math. 94, Amer. Math. Soc. (2017) 53 | DOI

[15] M Brion, Notes on automorphism groups of projective varieties, lecture notes (2018)

[16] M Brion, On models of algebraic group actions, Proc. Indian Acad. Sci. Math. Sci. 132 (2022) 61 | DOI

[17] C W Curtis, I Reiner, Representation theory of finite groups and associative algebras, 11, Wiley (1962) | DOI

[18] M R Darafsheh, Order of elements in the groups related to the general linear group, Finite Fields Appl. 11 (2005) 738 | DOI

[19] I Dolgachev, A Duncan, Automorphisms of cubic surfaces in positive characteristic, Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019) 15 | DOI

[20] I V Dolgachev, V A Iskovskikh, Finite subgroups of the plane Cremona group, from: "Algebra, arithmetic, and geometry : in honor of Yu I Manin, I" (editors Y Tschinkel, Y Zarhin), Progr. Math. 269, Birkhäuser (2009) 443 | DOI

[21] I Dolgachev, G Martin, Numerically trivial automorphisms of Enriques surfaces in characteristic 2, J. Math. Soc. Japan 71 (2019) 1181 | DOI

[22] B Fantechi, L Göttsche, L Illusie, S L Kleiman, N Nitsure, A Vistoli, Fundamental algebraic geometry : Grothendieck’s FGA explained, 123, Amer. Math. Soc. (2005) | DOI

[23] W Feit, Finite linear groups and theorems of Minkowski and Schur, Proc. Amer. Math. Soc. 125 (1997) 1259 | DOI

[24] G Frobenius, Über den von L Bieberbach gefundenen Beweis eines Satzes von C Jordan, Berl. Ber. 1911 (1911) 241

[25] A Grothendieck, Éléments de géométrie algébrique, IV : Étude locale des schémas et des morphismes de schémas, III, Inst. Hautes Études Sci. Publ. Math. 28 (1966) 5

[26] Y Gu, Surface fibrations with large equivariant automorphism group, preprint (2020)

[27] A Guld, Finite subgroups of the birational automorphism group are “almost” nilpotent of class at most two, preprint (2020)

[28] C D Hacon, C Xu, On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015) 711 | DOI

[29] M Hanamura, On the birational automorphism groups of algebraic varieties, Compos. Math. 63 (1987) 123

[30] M Hanamura, Structure of birational automorphism groups, I : Nonuniruled varieties, Invent. Math. 93 (1988) 383 | DOI

[31] R Hartshorne, Algebraic geometry, 52, Springer (1977) | DOI

[32] F Hu, Jordan property for algebraic groups and automorphism groups of projective varieties in arbitrary characteristic, Indiana Univ. Math. J. 69 (2020) 2493 | DOI

[33] S Iitaka, Algebraic geometry: an introduction to birational geometry of algebraic varieties, 24, Springer (1982)

[34] I M Isaacs, Finite group theory, 92, Amer. Math. Soc. (2008) | DOI

[35] V A Iskovskih, Minimal models of rational surfaces over arbitrary fields, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 19

[36] M C Jordan, Mémoire sur les équations différentielles linéaires à intégrale algébrique, J. Reine Angew. Math. 84 (1878) 89 | DOI

[37] J Keum, Orders of automorphisms of K3 surfaces, Adv. Math. 303 (2016) 39 | DOI

[38] C S H King, Generation of finite simple groups by an involution and an element of prime order, J. Algebra 478 (2017) 153 | DOI

[39] S L Kleiman, Toward a numerical theory of ampleness, Ann. of Math. 84 (1966) 293 | DOI

[40] S Kleiman, Les théorèmes de finitude pour le foncteur de Picard, from: "Théorie des intersections et théorème de Riemann–Roch (SGA 6)" (editors A Grothendieck, P Berthelot, L Illusie), Lecture Notes in Math. 225, Springer (1971) 616

[41] J Kollár, Rational curves on algebraic varieties, 32, Springer (1996) | DOI

[42] M J Larsen, R Pink, Finite subgroups of algebraic groups, J. Amer. Math. Soc. 24 (2011) 1105 | DOI

[43] R Lazarsfeld, Positivity in algebraic geometry, I : Classical setting : line bundles and linear series, 48, Springer (2004) | DOI

[44] C Liedtke, Algebraic surfaces in positive characteristic, from: "Birational geometry, rational curves, and arithmetic" (editors F Bogomolov, B Hassett, Y Tschinkel), Springer (2013) 229 | DOI

[45] J Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969) 195 | DOI

[46] J Lipman, Desingularization of two-dimensional schemes, Ann. of Math. 107 (1978) 151 | DOI

[47] Q Liu, Algebraic geometry and arithmetic curves, 6, Oxford Univ. Press (2002)

[48] Y I Manin, Cubic forms: algebra, geometry, arithmetic, 4, North-Holland (1986)

[49] M Martin-Deschamps, R Lewin-Ménégaux, Applications rationnelles séparables dominantes sur une variété de type général, Bull. Soc. Math. France 106 (1978) 279

[50] G Martin, Automorphism group schemes of bielliptic and quasi-bielliptic surfaces, Épijournal Géom. Algébrique 6 (2022) 9 | DOI

[51] H Matsumura, On algebraic groups of birational transformations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 34 (1963) 151

[52] S Meng, F Perroni, D Q Zhang, Jordan property for automorphism groups of compact spaces in Fujiki’s class C, J. Topol. 15 (2022) 806 | DOI

[53] S Meng, D Q Zhang, Jordan property for non-linear algebraic groups and projective varieties, Amer. J. Math. 140 (2018) 1133 | DOI

[54] H Minkowski, Zur Theorie der positiven quadratischen Formen, J. Reine Angew. Math. 101 (1887) 196 | DOI

[55] S Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982) 133 | DOI

[56] D Mumford, Enriques’ classification of surfaces in char p, I, from: "Global analysis" (editors D C Spencer, S Iyanaga), Univ. Tokyo Press (1969) 325 | DOI

[57] D Mumford, Abelian varieties, 5, Tata Inst. Fund. Res. (2008)

[58] I Mundet I Riera, Jordan’s theorem for the diffeomorphism group of some manifolds, Proc. Amer. Math. Soc. 138 (2010) 2253 | DOI

[59] A Ogus, Supersingular K3 crystals, from: "Journées de Géométrie Algébrique de Rennes, II", Astérisque 64, Soc. Math. France (1979) 3

[60] V L Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, from: "Affine algebraic geometry" (editors D Daigle, R Ganong, M Koras), CRM Proc. Lecture Notes 54, Amer. Math. Soc. (2011) 289 | DOI

[61] V L Popov, Birational splitting and algebraic group actions, Eur. J. Math. 2 (2016) 283 | DOI

[62] Y Prokhorov, C Shramov, Jordan property for groups of birational selfmaps, Compos. Math. 150 (2014) 2054 | DOI

[63] Y Prokhorov, C Shramov, Jordan property for Cremona groups, Amer. J. Math. 138 (2016) 403 | DOI

[64] Y Prokhorov, C Shramov, Jordan constant for Cremona group of rank 3, Mosc. Math. J. 17 (2017) 457 | DOI

[65] Y G Prokhorov, C A Shramov, Bounded automorphism groups of compact complex surfaces, Mat. Sb. 211 (2020) 105 | DOI

[66] Y Prokhorov, C Shramov, Automorphism groups of compact complex surfaces, Int. Math. Res. Not. 2021 (2021) 10490 | DOI

[67] Y G Prokhorov, K A Shramov, Automorphism groups of Moishezon threefolds, Mat. Zametki 106 (2019) 636 | DOI

[68] G R Robinson, On linear groups, J. Algebra 131 (1990) 527 | DOI

[69] J P Serre, Bounds for the orders of the finite subgroups of G(k), from: "Group representation theory" (editors M Geck, D Testerman, J Thévenaz), EPFL Press (2007) 405

[70] J P Serre, A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field, Mosc. Math. J. 9 (2009) 193 | DOI

[71] J P Serre, Finite groups: an introduction, 10, International (2016)

[72] I R Shafarevich, Basic algebraic geometry, 213, Springer (1974) | DOI

[73] C Shramov, V Vologodsky, Automorphisms of pointless surfaces, preprint (2018)

[74] C Shramov, V Vologodsky, Boundedness for finite subgroups of linear algebraic groups, Trans. Amer. Math. Soc. 374 (2021) 9029 | DOI

[75] J H Silverman, The arithmetic of elliptic curves, 106, Springer (1986) | DOI

[76] T A Springer, Linear algebraic groups, 9, Birkhäuser (1998) | DOI

[77] , The Stacks project, electronic reference (2005–)

[78] H Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik, I : Eine Abschätzung der Ordnung der Automorphismengruppe, Arch. Math. (Basel) 24 (1973) 527 | DOI

[79] K I Tahara, On the finite subgroups of GL(3, Z), Nagoya Math. J. 41 (1971) 169 | DOI

[80] T Tao, A proof of Gromov’s theorem, blog post (2010)

[81] K Ueno, Classification theory of algebraic varieties and compact complex spaces, 439, Springer (1975) | DOI

[82] W C Waterhouse, Two generators for the general linear groups over finite fields, Linear Multilinear Algebra 24 (1989) 227 | DOI

[83] A Weil, On algebraic groups of transformations, Amer. J. Math. 77 (1955) 355 | DOI

[84] R A Wilson, The finite simple groups, 251, Springer (2009) | DOI

[85] E Yasinsky, The Jordan constant for Cremona group of rank 2, Bull. Korean Math. Soc. 54 (2017) 1859 | DOI

[86] Y G Zarhin, Theta groups and products of abelian and rational varieties, Proc. Edinb. Math. Soc. 57 (2014) 299 | DOI