Rational surfaces with a large group of automorphisms

Cantat, Serge; Dolgachev, Igor

doi:10.1090/S0894-0347-2012-00732-2

Rational surfaces with a large group of automorphisms
HTML articles powered by AMS MathViewer

by Serge Cantat and Igor Dolgachev;

J. Amer. Math. Soc. 25 (2012), 863-905

DOI: https://doi.org/10.1090/S0894-0347-2012-00732-2

Published electronically: February 23, 2012

PDF

Abstract:

We classify rational surfaces $X$ for which the image of the automorphism group $\mathrm {Aut}(X)$ in the group of linear transformations of the Picard group $\mathrm {Pic}(X)$ is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in terms of periodic orbits of birational actions of infinite Coxeter groups.

References

Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225, DOI 10.1007/978-3-642-57739-0
W. Barth and C. Peters, Automorphisms of Enriques surfaces, Invent. Math. 73 (1983), no. 3, 383–411. MR 718937, DOI 10.1007/BF01388435
Eric Bedford and Kyounghee Kim, Periodicities in linear fractional recurrences: degree growth of birational surface maps, Michigan Math. J. 54 (2006), no. 3, 647–670. MR 2280499, DOI 10.1307/mmj/1163789919
Yves Benoist and Pierre de la Harpe, Adhérence de Zariski des groupes de Coxeter, Compos. Math. 140 (2004), no. 5, 1357–1366 (French, with English summary). MR 2081159, DOI 10.1112/S0010437X04000338
Serge Cantat, Dynamique des automorphismes des surfaces $K3$, Acta Math. 187 (2001), no. 1, 1–57 (French). MR 1864630, DOI 10.1007/BF02392831
S. Cantat, Sur la dynamique du groupe d’automorphismes des surfaces $K3$, Transform. Groups 6 (2001), no. 3, 201–214 (French, with English summary). MR 1854708, DOI 10.1007/BF01263089
Serge Cantat, Bers and Hénon, Painlevé and Schrödinger, Duke Math. J. 149 (2009), no. 3, 411–460. MR 2553877, DOI 10.1215/00127094-2009-042
Serge Cantat and Charles Favre, Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math. 561 (2003), 199–235 (French, with English summary). MR 1998612, DOI 10.1515/crll.2003.066
Arthur B. Coble, The Ten Nodes of the Rational Sextic and of the Cayley Symmetroid, Amer. J. Math. 41 (1919), no. 4, 243–265. MR 1506391, DOI 10.2307/2370285
Arthur B. Coble, Algebraic geometry and theta functions, American Mathematical Society Colloquium Publications, Vol. 10, American Mathematical Society, Providence, RI, 1982. Reprint of the 1929 edition. MR 733252
F. Cossec and I. Dolgachev, Smooth rational curves on Enriques surfaces, Math. Ann. 272 (1985), no. 3, 369–384. MR 799668, DOI 10.1007/BF01455565
François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969, DOI 10.1007/978-1-4612-3696-2
Michel Demazure, Henry Charles Pinkham, and Bernard Teissier (eds.), Séminaire sur les Singularités des Surfaces, Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980 (French). Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977. MR 579026
Vinay V. Deodhar, A note on subgroups generated by reflections in Coxeter groups, Arch. Math. (Basel) 53 (1989), no. 6, 543–546. MR 1023969, DOI 10.1007/BF01199813
I. Dolgachev, On automorphisms of Enriques surfaces, Invent. Math. 76 (1984), no. 1, 163–177. MR 739632, DOI 10.1007/BF01388499
I. Dolgachev, Infinite Coxeter groups and automorphisms of algebraic surfaces, The Lefschetz centennial conference, Part I (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1986, pp. 91–106. MR 860406, DOI 10.1090/conm/058.1/860406
Igor V. Dolgachev, Reflection groups in algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 1–60. MR 2358376, DOI 10.1090/S0273-0979-07-01190-1
Igor V. Dolgachev. Classical Algebraic Geometry: A Modern View, Cambridge Univ. Press, 2012, to appear.
Igor V. Dolgachev. Cremona special sets of points in products of projective spaces. (English) In Ebeling, Wolfgang (ed.) et al., Complex and differential geometry. Conference held at Leibniz Universität Hannover, Germany, September 14–18, 2009. Proceedings. Berlin: Springer. Springer Proceedings in Mathematics 8, 115–134 (2011).
Igor Dolgachev and David Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988), 210 pp. (1989) (English, with French summary). MR 1007155
Igor V. Dolgachev and De-Qi Zhang, Coble rational surfaces, Amer. J. Math. 123 (2001), no. 1, 79–114. MR 1827278
Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
M. H. Gizatullin, Rational $G$-surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 110–144, 239 (Russian). MR 563788
Brian Harbourne, Automorphisms of $K3$-like rational surfaces, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 17–28. MR 927971, DOI 10.1090/pspum/046.2/927971
Brian Harbourne, Rational surfaces with infinite automorphism group and no antipluricanonical curve, Proc. Amer. Math. Soc. 99 (1987), no. 3, 409–414. MR 875372, DOI 10.1090/S0002-9939-1987-0875372-8
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 463157
Seligmann Kantor. Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene. Berlin, Mayer & Müller, 1897.
Martin Kneser, Quadratische Formen, Springer-Verlag, Berlin, 2002 (German). Revised and edited in collaboration with Rudolf Scharlau. MR 2788987, DOI 10.1007/978-3-642-56380-5
Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni. MR 1096299, DOI 10.1007/978-3-642-75401-2
Max Lieblich and Davesh Maulik. On the cone conjecture for K$3$-surfaces in positive characteristic. arXiv:1102.3377, pages 1–8, 2011.
Curtis T. McMullen, Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 49–89. MR 2354205, DOI 10.1007/s10240-007-0004-x
D. R. Morrison, On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105–121. MR 728142, DOI 10.1007/BF01403093
David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, NJ, 1966. With a section by G. M. Bergman. MR 209285
Masayoshi Nagata, On rational surfaces. I. Irreducible curves of arithmetic genus $0$ or $1$, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1960), 351–370. MR 126443, DOI 10.1215/kjm/1250776405
Masayoshi Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 271–293. MR 126444, DOI 10.1215/kjm/1250775912
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
Viacheslav V. Nikulin. Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by $2$-reflections. algebro-geometric applications. J. Soviet Mathematics, 22:1401–1475, 1983.
Viacheslav V. Nikulin. Elliptic fibrations on K$3$ surfaces. arXiv:1010.3904, pages 1–19, 2012.
Jean-Pierre Serre, Cours d’arithmétique, Le Mathématicien [The Mathematician], No. 2, Presses Universitaires de France, Paris, 1977 (French). Deuxième édition revue et corrigée. MR 498338
Tomoyuki Takenawa, Algebraic entropy and the space of initial values for discrete dynamical systems, J. Phys. A 34 (2001), no. 48, 10533–10545. Symmetries and integrability of difference equations (Tokyo, 2000). MR 1877473, DOI 10.1088/0305-4470/34/48/317