Jordan Property for the Cremona Group over a Finite Field

Yuri G. Prokhorov ; Constantin A. Shramov

Informatics and Automation, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 298-310.

Résumé

We show that the Cremona group of rank $2$ over a finite field is Jordan, and provide an upper bound for its Jordan constant which is sharp when the number of elements in the field is different from $2$, $4$, and $8$.

Bibliographie

[1] Bădescu L., Algebraic surfaces, Universitext, Springer, New York, 2001 | DOI | MR

[2] Brauer R., Feit W., “An analogue of Jordan's theorem in characteristic $p$”, Ann. Math. Ser. 2, 84 (1966), 119–131 | DOI | MR | Zbl

[3] Cantat S., “Birational permutations”, C. r. Math. Acad. sci. Paris, 347:21–22 (2009), 1289–1294 | DOI | MR | Zbl

[4] Chen Y., Shramov C., Automorphisms of surfaces over fields of positive characteristic, E-print, 2021, arXiv: 2106.15906

[5] Dolgachev I.V., Classical algebraic geometry: A modern view, Cambridge Univ. Press, Cambridge, 2012 | MR | Zbl

[6] I. Dolgachev and A. Duncan, “Automorphisms of cubic surfaces in positive characteristic”, Izv. Math., 83:3 (2019), 424–500 | DOI | DOI | MR | Zbl

[7] Hu F., “Jordan property for algebraic groups and automorphism groups of projective varieties in arbitrary characteristic”, Indiana Univ. Math. J., 69:7 (2020), 2493–2504 | DOI | MR | Zbl

[8] V. A. Iskovskih, “Minimal models of rational surfaces over arbitrary fields”, Math. USSR, Izv., 14:1 (1980), 17–39 | DOI | MR | Zbl

[9] Jacobson N., Lectures in abstract algebra. III: Theory of fields and Galois theory, Grad. Texts Math., 32, Springer, New York, 1975 | MR

[10] Jordan C., “Mémoire sur les équations différentielles linéaires à intégrale algébrique”, J. reine angew. Math., 84 (1877), 89–215 | MR

[11] Kollár J., Rational curves on algebraic varieties, Ergebn. Math. Grenzgeb. 3. Folge, 32, Springer, Berlin, 1996 | MR

[12] Kollár J., Severi–Brauer varieties; a geometric treatment, E-print, 2016, arXiv: 1606.04368

[13] Kuznetsova A., “Automorphisms of quasi-projective surfaces over fields of finite characteristic”, J. Algebra, 595 (2022), 271–278 | DOI | MR | Zbl

[14] Larsen M.J., Pink R., “Finite subgroups of algebraic groups”, J. Amer. Math. Soc., 24:4 (2011), 1105–1158 | DOI | MR | Zbl

[15] Mori S., “Threefolds whose canonical bundles are not numerically effective”, Ann. Math. Ser. 2, 116:1 (1982), 133–176 | DOI | MR | Zbl

[16] Popov V.L., “On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties”, Affine algebraic geometry: The Russell Festschrift, CRM Proc. Lect. Notes, 54, Amer. Math. Soc., Providence, RI, 2011, 289–311 | DOI | MR | Zbl

[17] Popov V.L., “Jordan groups and automorphism groups of algebraic varieties”, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., 79, Springer, Cham, 2014, 185–213 | MR | Zbl

[18] Yu. G. Prokhorov, “Equivariant minimal model program”, Russ. Math. Surv., 76:3 (2021), 461–542 | DOI | DOI | MR | Zbl

[19] J.-P. Serre, A Course in Arithmetic, Grad. Texts Math., 7, Springer, New York, 1973 | DOI | MR | MR | Zbl

[20] Serre J.-P., “A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field”, Moscow Math. J., 9:1 (2009), 183–198 | DOI | MR | Zbl

[21] Shramov C., Vologodsky V., Automorphisms of pointless surfaces, E-print, 2018, arXiv: 1807.06477

[22] Stichtenoth H., “Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I: Eine Abschätzung der Ordnung der Automorphismengruppe”, Arch. Math., 24 (1973), 527–544 | DOI | MR | Zbl

[23] Várilly-Alvarado A., “Arithmetic of del Pezzo surfaces”, Birational geometry, rational curves, and arithmetic, Simons Symp., Springer, New York, 2013, 293–319 | MR | Zbl

[24] Wilson R.A., The finite simple groups, Grad. Texts Math., 251, Springer, London, 2009 | DOI | MR | Zbl

[25] Yasinsky E., “The Jordan constant for Cremona group of rank 2”, Bull. Korean Math. Soc., 54:5 (2017), 1859–1871 | MR | Zbl