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Transactions of the American Mathematical Society

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Plane Cremona maps, exceptional curves and roots

Author: Maria Alberich-Carramiñana
Journal: Trans. Amer. Math. Soc. 357 (2005), 1901-1914
MSC (2000): Primary 14J26, 14E05, 14E07
Published electronically: December 10, 2004
MathSciNet review: 2115081
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Abstract: We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection $-1$ and $-2$, respectively) on a smooth complex projective surface $S$ which admits a birational morphism $\pi$ to $\mathbb{P} ^{2}$. The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of $C$ is in the $W$-orbit of the class of the total transform of some point blown up by $ \pi $ if $C$ is exceptional, or in the $W$-orbit of a simple root if $C$ is root, where $W$ is the Weyl group acting on $\operatorname{Pic}S$; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor $C$ is a necessary and sufficient condition in order that $\pi_{\ast} (C)$ could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.

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  • 1. Maria Alberich-Carramiñana, Geometry of the plane Cremona maps, Lecture Notes in Mathematics, vol. 1769, Springer-Verlag, Berlin, 2002. MR 1874328
  • 2. Eduardo Casas-Alvero, Singularities of plane curves, London Mathematical Society Lecture Note Series, vol. 276, Cambridge University Press, Cambridge, 2000. MR 1782072
  • 3. 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
  • 4. 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
  • 5. P. Du Val, On the Kantor group of a set of points in a plane, Proc. London Math. Soc. 42 (1936), no. 2, 18-51.
  • 6. F. Enriques and O. Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche, N. Zanichelli, Bologna, 1915.
  • 7. Brian Harbourne, Blowings-up of 𝑃² and their blowings-down, Duke Math. J. 52 (1985), no. 1, 129–148. MR 791295,
  • 8. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • 9. H.P. Hudson, Cremona transformations in plane and space, Cambridge University Press, 1927.
  • 10. M. Lahyane, Irreducibility of the $(-1)$-classes on smooth rational surfaces, Preprint IC2001098P of The Abdus Salam International Centre for Theoretical Physics (2001).
  • 11. Eduard Looijenga, Rational surfaces with an anticanonical cycle, Ann. of Math. (2) 114 (1981), no. 2, 267–322. MR 632841,
  • 12. Yu. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR 833513
  • 13. Masayoshi Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1960/1961), 271–293. MR 0126444
  • 14. I. R. Shafarevich, Algebraic surfaces, Proceedings of the Steklov Institute of Mathematics, vol. 75, American Mathematical Society, 1967.

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Additional Information

Maria Alberich-Carramiñana
Affiliation: Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028-Barcelona, Spain

Received by editor(s): April 11, 2003
Received by editor(s) in revised form: August 22, 2003
Published electronically: December 10, 2004
Additional Notes: The author completed this work as a researcher of the Programa Ramón y Cajal of the Ministerio de Ciencia y Tecnología, and was also supported in part by CAICYT BFM2002-012040, Generalitat de Catalunya 2000SGR-00028 and EAGER, European Union contract HPRN-CT-2000-00099
Article copyright: © Copyright 2004 American Mathematical Society