Hostname: page-component-6bf8c574d5-k2jvg Total loading time: 0 Render date: 2025-03-07T03:08:43.979Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite quasisimple groups acting on rationally connected threefolds

Part of: Surfaces and higher-dimensional varieties Representation theory of groups Permutation groups Algebraic groups Birational geometry

Published online by Cambridge University Press:  06 December 2022

JÉRÉMY BLANC ,
IVAN CHELTSOV ,
ALEXANDER DUNCAN  and
YURI PROKHOROV
JÉRÉMY BLANC
Affiliation:
Departement Mathematik und Informatik Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland. e-mail: jeremy.blanc@unibas.ch
IVAN CHELTSOV
Affiliation:
School of Mathematics, The University of Edinburgh, Edinburgh EH9 and 3JZ. e-mail: I.Cheltsov@ed.ac.uk
ALEXANDER DUNCAN
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. e-mail: duncan@math.sc.edu
YURI PROKHOROV
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkina str., Moscow 119991, Russia and AG Laboratory, National Research University Higher School of Economics, 6 Usacheva str., Moscow, 119048, Russia. e-mail: prokhoro@mi-ras.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , ${\mathfrak{A}}_7$ , ${\text{PSp}}_4(\textbf{F}_3)$ , ${\text{SL}}_2(\textbf{F}_{7})$ , $2.{\mathfrak{A}}_5$ , $2.{\mathfrak{A}}_6$ , $3.{\mathfrak{A}}_6$ or $6.{\mathfrak{A}}_6$ . All of these groups with a possible exception of $2.{\mathfrak{A}}_6$ and $6.{\mathfrak{A}}_6$ indeed act on some rationally connected threefolds.

MSC classification

Primary: 14E07: Birational automorphisms, Cremona group and generalizations 14L30: Group actions on varieties or schemes (quotients) 14J45: Fano varieties
Secondary: 20D05: Finite simple groups and their classification 20D99: None of the above, but in this section 14J30: $3$-folds 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 3 , May 2023 , pp. 531 - 568
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arezzo, C., Ghigi, A., and Pirola, G. P.. Symmetries, quotients and Kähler–Einstein metrics. J. Reine Angew. Math. 591 (2006), 177–200.Google Scholar
Aschbacher, M.. Finite Group Theory. Cambridge Studies in Advanced Math. no. 10 (Cambridge University Press, Cambridge, second edition, 2000).Google Scholar
Avilov, A. A.. Existence of standard models of conic fibrations over non-algebraically-closed fields. Sb. Math. 205(12) (2014), 1683–1695.CrossRefGoogle Scholar
Białynicki–Birula, A.. Some theorems on actions of algebraic groups. Ann. of Math. (2) 98 (1973), 480–497.Google Scholar
Blanc, J.. Elements and cyclic subgroups of finite order of the Cremona group. Comment. Math. Helv. 86(2) (2011), 469–497.Google Scholar
Blichfeldt, H. F.. Finite Collineation Groups (The University of Chicago Press, Chicago, 1917).Google Scholar
Cartan, H.. Quotient d’un espace analytique par un groupe d’automorphismes. In Algebraic Geometry and Topology. (Princeton University Press, Princeton, N. J., 1957), pp. 90–102. A symposium in honor of S. Lefschetz.Google Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.. Atlas of Finite Groups (Oxford University Press, Eynsham, 1985). Maximal subgroups and ordinary characters for simple groups, with computational assistance from J. G. Thackray.Google Scholar
Colliot–Thélène, J.-L., Karpenko, N. A. and Merkur’ev, A. S.. Rational surfaces and the canonical dimension of $\text{PGL}_6$ . St. Petersburg. Math. J. 19(5) (2008), 793–804.Google Scholar
Cohen, A. M.. Finite complex reflection groups. Ann. Sci. École Norm. Sup. (4), 9(3) (1976), 379–436.Google Scholar
Corti, A.. Singularities of linear systems and 3-fold birational geometry. In Explicit Birational Geometry of 3-folds. London Math. Soc. Lecture Note Ser. vol. 281 (Cambridge Univ. Press, Cambridge, 2000), pp. 259–312.Google Scholar
Cox, D. A.. The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1) (1995), 17–50.Google Scholar
Cheltsov, I., Przyjalkowski, V. and Shramov, C.. Burkhardt quartic, Barth sextic, and the icosahedron. Internat. Math. Res. Notices. 2019(12) (2019), 3683–3703.Google Scholar
Cheltsov, I. A. and Shramov, K. A.. Log-canonical thresholds for nonsingular Fano threefolds. Uspekhi Mat. Nauk. 63(5(383)) (2008), 73–180.Google Scholar
I. Cheltsov and C. Shramov. Three embeddings of the Klein simple group into the Cremona group of rank three. Transform. Groups. 17(2) (2012), 303–350.CrossRefGoogle Scholar
I. Cheltsov and C. Shramov. Five embeddings of one simple group. Trans. Amer. Math. Soc. 366(3) (2014).Google Scholar
I. Cheltsov and C. Shramov. Cremona Groups and the Icosahedron. Boca Raton, FL (CRC Press, Boca Raton, Fl., 2016.Google Scholar
I. Cheltsov and C. Shramov. Finite collineation groups and birational rigidity. Arxiv e-print, 1712.08258 (2017).Google Scholar
Dolgachev, I. V. and Iskovskikh, V. A.. Finite subgroups of the plane Cremona group. In Algebra, Arithmetic and geometry: in honor of Yu. I. Manin. Vol. I. Progr. Math. vol. 269. (Birkhäuser Boston Inc., Boston, MA, (2009), pp. 443–548.Google Scholar
Dolgachev, I. V.. Invariant stable bundles over modular curves X(p). In Recent progress in algebra (Taejon/Seoul, 1997), Contemp. Math. vol. 224 (Amer. Math. Soc., Providence, RI, 1999), pp. 65–99.Google Scholar
Edge, W. L.. The Klein group in three dimensions. Acta Math. 79 (1947), 153–223.Google Scholar
D. Eisenbud and J. Harris. On varieties of minimal degree. (A centennial account). In Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), part 1, Proc. Sympos. Pure Math. vol. 46 (Amer. Math. Soc.,Providence, RI, 1987), pp. 3–13.Google Scholar
Fujita, T.. On singular del Pezzo varieties. In Algebraic geometry (L’Aquila, 1988). Lecture Notes in Math. vol. 1417 (Springer, Berlin,1990), pp. 117–128.Google Scholar
The GAP Group. GAP – Groups, Algorithms and Programming, version 4.9.1 (2018), 2018.Google Scholar
Iskovskikh, V. A. and Manin, Y. I.. Three-dimensional quartics and counterexamples to the Lüroth problem. Math. USSR-Sb. 15 (1971), 141–166.Google Scholar
Iskovskikh, V. A. and Prokhorov, Y.. Fano Varieties. Algebraic Geometry V. Encyclopaedia Math. Sci. vol. 47 (Springer, Berlin, 1999).Google Scholar
Iskovskikh, V. A.. Anticanonical models of three-dimensional algebraic varieties. J. Sov. Math. 13 (1980), 745–814.Google Scholar
Iskovskikh, V. A.. Birational automorphisms of three-dimensional algebraic varieties. J. Sov. Math. 13 (1980), 815–868.Google Scholar
Kawamata, Y.. Boundedness of $\textbf{Q}$ -Fano threefolds. In Proceedings of the International Conference on Algebra, part 3 (Novosibirsk, 1989) Contemp. Math. vol. 131 ( Amer. Math. Soc., Providence, RI, 1992), pp. 439–445.Google Scholar
Kawamata, Y.. On Fujita’s freeness conjecture for 3-folds and 4-folds. Math. Ann. 308(3) (1997), 491–505.CrossRefGoogle Scholar
Kawamata, Y.. Subadjunction of log canonical divisors. II. Amer. J. Math. 120(5) (1998), 893–899.CrossRefGoogle Scholar
Kawakita, M.. Inversion of adjunction on log canonicity. Invent. Math. 167 (2007), 129–133.CrossRefGoogle Scholar
J. Kollár and S. Mori. Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics vol. 134 (Cambridge University Press, Cambridge, 1998). With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original.Google Scholar
J. Kollár, Y. Miyaoka, Mori, S. and Takagi, H.. Boundedness of canonical $\textbf{Q}$ -Fano 3-folds. Proc. Japan Acad. Ser. A Math. Sci. 76(5) (2000), 73–77.CrossRefGoogle Scholar
Kollár, J., editor. Flips and abundance for algebraic threefolds (Société Mathématique de France, Paris, 1992). Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah (August 1991). Astérisque 211 (1992).Google Scholar
Kollár, J.. Singularities of pairs. In Algebraic geometry–Santa Cruz 1995. Proc. Sympos. Pure Math. vol. 62 (Amer. Math. Soc., Providence, RI, 1997), pp. 221–287.CrossRefGoogle Scholar
Krylov, I.. Birational geometry of del Pezzo fibrations with terminal quotient singularities. J. London Math. Soc., 2018, 97, 222–246.Google Scholar
Kollár, J. and Shepherd–Barron, N. I.. Threefolds and deformations of surface singularities. Invent. Math. 91(2) (1988), 299–338.Google Scholar
Lazarsfeld, R.. Positivity in algebraic geometry., II. Ergeb. Math. Grenzgeb. 3, vol. 49. Folge. A Series of Modern Surveys in Mathematics (Springer-Verlag, Berlin, 2004). Positivity for vector bundles, and multiplier ideals.Google Scholar
Liedtke, C.. Morphisms to Brauer–Severi varieties, with applications to del Pezzo surfaces. In Geometry Over Nonclosed Fields, Simons Symp. (Springer, Cham, 2017), pp. 157–196.Google Scholar
Liu, Y.. K-stability of cubic fourfolds. J. Reine Angew. Math. 786 (2022), 55–77.Google Scholar
Mallows, C. L. and Sloane, N. J. A.. On the invariants of a linear group of order 336. Proc. Camb. Phil. Soc. 74 (1973), 435–440.Google Scholar
Mukai, S.. Finite groups of automorphisms of K3 surfaces and the Mathieu group. Invent. Math. 94(1) (1988), 183–221.Google Scholar
Namikawa, Y.. Smoothing Fano 3-folds. J. Algebraic Geom. 6(2) (1997), 307–324.Google Scholar
Przhiyalkovskij, V. V., Chel’tsov, I. A. and Shramov, K. A.. Hyperelliptic and trigonal Fano threefolds. Izv. Math. 69(2) (2005), 365–421.Google Scholar
Prokhorov, Y.. Simple finite subgroups of the Cremona group of rank 3. J. Algebraic Geom. 21(3) (2012), 563–600.CrossRefGoogle Scholar
Prokhorov, Y.. G-Fano threefolds, I. Adv. Geom. 13(3) (2013), 389–418.Google Scholar
Prokhorov, Y.. G-Fano threefolds, II. Adv. Geom. 13(3) (2013), 419–434.Google Scholar
Prokhorov, Y.. Singular Fano threefolds of genus 12. Sb. Math. 207(7) (2016), 983–1009.Google Scholar
Prokhorov, Y.. The rationality problem for conic bundles. Russian Math. Surv. 73(3) 2018.Google Scholar
Reid, M.. Young Person’s Guide to Canonical Singularities. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math. vol. 46 (Amer. Math. Soc., Providence, RI, 1987), pp. 345–414.Google Scholar
Reichstein, Z. and Youssin, B.. Equivariant resolution of points of indeterminacy. Proc. Amer. Math. Soc. 130(8) (2002), 2183–2187.CrossRefGoogle Scholar
Sarkisov, V. G.. On the structure of conic bundles. Math. USSR, Izv. 120 (1982), 355–390.Google Scholar
Shin, K.-H.. 3-dimensional Fano varieties with canonical singularities. Tokyo J. Math. 12(2) (1989), 375–385.Google Scholar
Shokurov, V. V.. 3-fold log flips. Russ. Acad. Sci., Izv., Math. 40(1) (1993), 95–202.Google Scholar
Tian, G.. On Kähler–Einstein metrics on certain Kähler manifolds with $C_1(M)>0$ . Invent. Math. 89(2) (1987), 225–246.CrossRefGoogle Scholar
Tsygankov, V. I.. Equations of G-minimal conic bundles. Sb. Math., 202(11-12):1667–1721, 2011. Translated from Mat. Sb. 202(11) (2011), 103–160.Google Scholar
Tsygankov, V. I.. The conjugacy classes of finite nonsolvable subgroups in the plane Cremona group. Adv. Geom. 13(2) (2013), 323–347.Google Scholar
Wavrik, J.J.. Deformations of branched coverings of complex manifolds. Amer. J. Math. 90 (1968), 926–960.Google Scholar