Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. M. Alberich-Carramiñana, Geometry of the plane Cremona maps, Lecture Notes in Math., vol. 1769, Springer, Heidelberg, 2002. MR 1874328 (2002m:14008)
  • 2. E. Casas-Alvero, Singularities of plane curves, London Math. Soc. Lecture Note Ser., vol. 276, Cambridge University Press, 2000. MR 1782072 (2003b:14035)
  • 3. M. Demazure, Surfaces de del Pezzo I, II, III, IV, V, Lecture Notes in Math., vol. 777, Springer, Heidelberg, 1976. MR 0579026 (82d:14021)
  • 4. I. Dolgachev and D. Ortland, Point sets in projective spaces and theta functions, Astérisque, vol. 165, Soc. Math. France, Paris, 1988. MR 1007155 (90i:14009)
  • 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. B. Harbourne, Blowings-up of $\mathbb{P} ^{2}$ and their blowings-down, Duke Math. J. 52 (1985), 129-148. MR 0791295 (86m:14026)
  • 8. R. Hartshorne, Algebraic geometry, Grad. Texts in Math., vol. 52, Springer, New York, 1977. MR 0463157 (57:3116)
  • 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. E. Looijenga, Rational surfaces with an anti-canonical cycle, Ann. of Math. 114 (1981), 267-322. MR 0632841 (83j:14030)
  • 12. Y. I. Manin, Cubic forms, North-Holland Math. Library, vol. 4, North-Holland, Amsterdam, 1986. MR 0833513 (87d:11037)
  • 13. M. Nagata, On rational surfaces II, Memoirs of the College of Science, University of Kyoto, Series A 33 (1960), no. 2, 271-293. MR 0126444 (23:A3740)
  • 14. I. R. Shafarevich, Algebraic surfaces, Proceedings of the Steklov Institute of Mathematics, vol. 75, American Mathematical Society, 1967.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14J26, 14E05, 14E07

Retrieve articles in all journals with MSC (2000): 14J26, 14E05, 14E07

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

American Mathematical Society