Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T08:03:34.033Z Has data issue: false hasContentIssue false
  • English
  • Français

On planar Cremona maps of prime order

Published online by Cambridge University Press:  22 January 2016

Tommaso de Fernex
Tommaso de Fernex*
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, USA, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper contains a new proof of the classification of prime order elements of Bir(ℙ2) up to conjugation. The first results on this topic can be traced back to classic works by Bertini and Kantor, among others. The innovation introduced by this paper consists of explicit geometric constructions of these Cremona transformations and the parameterization of their conjugacy classes. The methods employed here are inspired to [4], and rely on the reduction of the problem to classifying prime order automorphisms of rational surfaces. This classification is completed by combining equivariant Mori theory to the analysis of the action on anticanonical rings, which leads to characterize the cases that occur by explicit equations (see [28] for a different approach). Analogous constructions in higher dimensions are also discussed.

Keywords

14E0714J5014E20
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 174 , 2004 , pp. 1 - 28
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[1] Alberich-Caraminana, M., Geometry of the Plane Cremona Maps, Lectures Notes in Math., 1769 (2002), Springer-Verlag, Berlin.Google Scholar
[2] Autonne, L., Reserches sur les groupes d’ordre fini contenus dans le groupe Cremona, J. Math. Pures et Appl. (1885).Google Scholar
[3] Barth, W., Peters, C. and Ven, A. Van de, Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 4, Springer-Verlag, Berlin, 1984.CrossRefGoogle Scholar
[4] Bayle, L. and Beauville, A., Birational involutions of ℙ2 , Asian J. Math., 4(1) (2000), 11–18.CrossRefGoogle Scholar
[5] Beauville, A., On Cremona transformations of prime order, Preprint available at ArXiv:math.AG/0402037.Google Scholar
[6] Bertini, E., Ricerche sulle trasformazioni univoche involutorie nel piano, Annali di Mat., 8 (1877), 244–286.CrossRefGoogle Scholar
[7] Bottari, A., Sulla razionalità dei piani multipli , Ann. Mat. Pura Appl., Serie III, 2 (1899), 277–296.Google Scholar
[8] Calabri, A., On rational and ruled double planes, Ann. Mat. Pura Appl., 181(4) (2002), 365–387.Google Scholar
[9] Calabri, A., On rational cyclic triple planes, Preprint (2001).Google Scholar
[10] Castelnuovo, G. and Enriques, F., Sulle condizioni di razionalità dei piani doppi, Rend. del Circ. Mat. di Palermo, 14 (1900), 290–302.CrossRefGoogle Scholar
[11] Fernex, T. de and Ein, L., Resolution of indeterminacy of pairs, in Algebraic Geometry, a Volume in Memory of Paolo Francia, eds. Beltrametti, M. et al. (2002), de Gruiter.Google Scholar
[12] Engel, W., Invariante Divisorensharen dei endlichen Gruppen von Cremonatransformationen, J. Reine Angew. Math., 196 (1956), 59–66.Google Scholar
[13] Fujita, T., Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, 155 (1990), Cambridge Univ. Press, Cambridge.Google Scholar
[14] Gizatullin, M. H., Rational G-surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 44(1) (1980), 110–144, 239.Google Scholar
[15] Godeaux, L., Une représentation des transformations birationnelles du plan et de l’espace, Acad. Roy. Belgique. Cl. Sci. Mém. Coll. in 8°. (2), 24(2) (1949), 31.Google Scholar
[16] Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977, pp. xvi+496.Google Scholar
[17] Hosoh, T., Automorphism groups of quartic del Pezzo surfaces, J. Algebra, 185(2) (1996), 374–389.Google Scholar
[18] Hosoh, T., Automorphism groups of cubic surfaces, J. Algebra, 192(2) (1997), 651–677.CrossRefGoogle Scholar
[19] Kantor, S., Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Mayer & Müller, Berlin, 1895.Google Scholar
[20] Koitabashi, M., Automorphism groups of generic rational surfaces, J. Algebra, 116(1) (1988), 130–142.Google Scholar
[21] Kollár, J. and Mori, S., Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and Corti A., Translated from the 1998 Japanese original, Cambridge University Press, Cambridge, 1998, pp. viii+254.Google Scholar
[22] Manin, Yu. I., Rational surfaces over perfect fields. II, Math. USSR Sb., 1 (1967), 141–168.CrossRefGoogle Scholar
[23] Manin, Yu. I., Cubic forms: algebra, geometry, arithmetic, Translated from Russian by Hazewinkel, M., North-Holland Mathematical Library, Vol. 4, North-Holland Publishing Co., Amsterdam, 1974, pp. vii+292.Google Scholar
[24] Miranda, R. and Persson, U., On extremal rational elliptic surfaces, Math. Z., 193(4) (1986), 537–558.Google Scholar
[25] Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2), 116(1) (1982), 133–176.CrossRefGoogle Scholar
[26] Segre, B., The non-singular cubic surface (1942), Oxford University Press, Oxford.Google Scholar
[27] Wiman, A., Zur theorie der endlichen gruppen von birazionalen transformationen in der ebene, Math. Ann., 4 (1897).Google Scholar
[28] Zhang, D.-Q., Automorphisms of finite order on rational surfaces, J. Algebra, 238(2) (2001), 560–589.Google Scholar
[29] Zhang, D.-Q., Automorphisms of finite order on Gorenstein del Pezzo surfaces, Trans. Amer. Math. Soc., 354(12) (2002), 4831–4845.Google Scholar