Summary
This paper completes the classic and modern results on classification of conjugacy classes of finite subgroups of the group of birational automorphisms of the complex projective plane.
Key words
2000 Mathematics Subject Classifications: 14E07 (Primary); 14J26, 14J50, 20B25 (Secondary)
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- 1.
The author was supported in part by NSF grant 0245203.
- 2.
The author was supported in part by RFBR 05-01-00353-a RFBR 08-01-00395-a, grant CRDF RUMI 2692-MO-05 and grant of NSh 1987-2008.1.
- 3.
We thank V. Tsygankov for this observation.
- 4.
We thank J. Blanc for pointing out a mistake in the statement of this theorem in an earlier version of this paper. The correct statement had appeared first in his paper [7].
- 5.
A better argument due to I. Cheltsov shows that in this and the previous cases B is conjugate to a group of automorphisms of \({\mathbb{P}}^2\), or \({\mathbf{F}}_0\), or \({\mathbf{F}}_2\).
References
A. Adem, J. Milgram, Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften, 309, Springer-Verlag, Berlin, 1994.
M. Alberich-CarramiÃČÂćÃćâĂŽÂňÃćâĆňÅŞana, Geometry of the plane Cremona maps, Lecture Notes in Mathematics, 1769. Springer-Verlag, Berlin, 2002.
L. Autonne, Recherches sur les groupes d'ordre fini contenus dans le groupe Cremona, Premier Mémoire Généralités et groupes quadratiques, J. Math. Pures et Appl., (4) 1 (1885), 431–454.
S. Bannai, H. Tokunaga, A note on embeddings of \(S_4\) and \(A_5\) into the Cremona group and versal Galois covers, Publ. Res. Inst. Math. Sci. 43 (2007), 1111–1123.
L. Bayle, A. Beauville, Birational involutions of \(P^2\), Asian J. Math. 4 (2000), 11–17.
J. Blanc, Finite abelian subgroups of the Cremona group of the plane, thesis, Univ. of Geneva, 2006.
J. Blanc, Elements and cyclic subgroups of finite order of the Cremona group, to appear in Com. Math. Helv.
A. Beauville, J. Blanc, On Cremona transformations of prime order, C. R. Math. Acad. Sci. Paris 339 (2004), 257–259.
A. Beauville, p-elementary subgroups of the Cremona group, J. Algebra, 314 (2007), 553–564.
E. Bertini, Ricerche sulle trasformazioni univoche involutorie nel piano, Annali di Mat. Pura Appl. (2) 8 (1877), 254–287.
H. Blichfeldt, Finite collineation groups, with an introduction to the theory of operators and substitution groups, Univ. of Chicago Press, Chicago, 1917.
A. Bottari, Sulla razionalità dei piani multipli \(\{x,y,\sqrt[n]{F(x,y)}\}\), Annali di Mat. Pura Appl. (3) 2 (1899), 277–296.
A. Calabri, Sulle razionalità dei piani doppi e tripli cyclici, Ph.D. thesis, Univ. di Roma “La Sapienza”, 1999.
A. Calabri, On rational and ruled double planes, Annali. di Mat. Pura Appl. (4) 181 (2002), 365–387.
R. Carter, Conjugacy classes in the Weyl group. in “Seminar on Algebraic Groups and Related Finite Groups”, The Institute for Advanced Study, Princeton, N.J., 1968/69, pp. 297–318, Springer, Berlin.
G. Castelnuovo, Sulle razionalità delle involutioni piani, Math. Ann, 44 (1894), 125–155.
G. Castelnuovo, F. Enriques, Sulle condizioni di razionalità dei piani doppia, Rend. Circ. Mat. di Palermo, 14 (1900), 290–302.
A. Coble, Algebraic geometry and theta functions (reprint of the 1929 edition), A.M.S. Coll. Publ., v. 10. A.M.S., Providence, RI, 1982. MR0733252 (84m.14001)
J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson, Atlas of finite groups, Oxford Univ. Press, Eynsham, 1985.
J. Conway, D. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd., Natick, MA, 2003.
J. Coolidge, A treatise on algebraic plane curves, Dover Publ. New York. 1959.
A. Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), 223–254.
T. de Fernex, On planar Cremona maps of prime order, Nagoya Math. J. 174 (2004), 1–28.
T. de Fernex, L. Ein, Resolution of indeterminacy of pairs, in “Algebraic geometry”, pp. 165–177, de Gruyter, Berlin, 2002.
M. Demazure, Surfaces de Del Pezzo, I–V, in “Séminaire sur les Singularités des Surfaces”, ed. by M. Demazure, H. Pinkham and B. Teissier. Lecture Notes in Mathematics, 777. Springer, Berlin, 1980, pp. 21–69.
I. Dolgachev, Weyl groups and Cremona transformations. in “Singularities, Part 1 (Arcata, Calif., 1981)”, 283–294, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983.
I. Dolgachev, Topics in classical algebraic geometry, Part I, manuscript in preparations, see www.math.lsa.umich.edu/idolga/lecturenotes.html.
P. Du Val, On the Kantor group of a set of points in a plane, Proc. London Math. Soc. 42 (1936), 18– 51.
F. Enriques, Sulle irrazionalita da cui puo farsi dipendere la resoluzione d'un' equazione algebrica \(f(x,y,z) = 0\) con funzioni razionali di due parametri, Math. Ann. 49 (1897), 1–23.
C. Geiser, Über zwei geometrische Probleme, J. Reine Angew. Math. 67 (1867), 78–89.
D. Gorenstein, Finite groups, Chelsea Publ. Co., New York, 1980.
É. Goursat, Sur les substitutions orthogonales et les divisions rÃČâĂęÃĆ¡guliÃČâĂŽÃĆÂŔres de l'espace, Ann. de e'Ècole Norm. Sup. (3) 6 (1889), 9–102.
R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
T. Hosoh, Automorphism groups of cubic surfaces, J. Algebra 192 (1997), 651–677.
H. Hudson, Cremona transformations in plane and space, Cambridge Univ. Press. 1927.
V. A. Iskovskih, Rational surfaces with a pencil of rational curves (Russian) Mat. Sb. (N.S.) 74 (1967), 608–638.
V. A. Iskovskikh, Rational surfaces with a pencil of rational curves with positive square of the canonical class, Math. USSR Sbornik, 12 (1970), 93–117.
V. A. Iskovskih, Minimal models of rational surfaces over arbitrary fields (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 19–43,
V. A. Iskovskikh, Factorization of birational mappings of rational surfaces from the point of view of Mori theory (Russian) Uspekhi Mat. Nauk 51 (1996), 3–72; translation in Russian Math. Surveys 51 (1996), 585–652.
V. A. Iskovskikh, Two nonconjugate embeddings of the group \(S_3\times Z_2\) into the Cremona group (Russian) Tr. Mat. Inst. Steklova 241 (2003), Teor. Chisel, Algebra i Algebr. Geom., 105–109; translation in Proc. Steklov Inst. Math. 241 (2003), 93–97.
V. A. Iskovskikh, Two non-conjugate embeddings of \(S_3\times {\mathbb{Z}}_2\) into the Cremona group II, Adv. Study in Pure Math. 50 (2008), 251–267.
S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Berlin. Mayer & MÃČâĂęÃĆÂÿller. 111 S. gr. \(8^\circ\). 1895.
J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.
N. Lemire, V. Popov, Z. Reichstein, Cayley groups, J. Amer. Math. Soc. 19 (2006), 921–967.
Yu. I. Manin, Rational surfaces over perfect fields. II (Russian) Mat. Sb. (N.S.) 72 (1967), 161–192.
Yu. I. Manin, Cubic forms: algebra, geometry, arithmetic Translated from Russian by M. Hazewinkel. North-Holland Mathematical Library, Vol. 4. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974.
M. Noether, Über die ein-zweideutigen Ebenentransformationen, Sitzungberichte der physic-medizin, Soc. zu Erlangen, 1878.
B. Segre, The non-singular cubic surface, Oxford Univ. Press. Oxford. 1942.
T. Springer, Invariant theory. Lecture Notes in Mathematics, Vol. 585. Springer-Verlag, Berlin-New York, 1977.
A. Wiman, Zur Theorie endlichen Gruppen von birationalen Transformationen in der Ebene, Math. Ann. 48 (1896), 195–240.
H. Zassenhaus, The theory of groups, Chelsea Publ. New York, 1949.
D.-Q. Zhang, Automorphisms of finite order on rational surfaces. With an appendix by I. Dolgachev, J. Algebra 238 (2001), 560–589.
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Dolgachev, I.V., Iskovskikh, V.A. (2009). Finite Subgroups of the Plane Cremona Group. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_11
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