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Transactions of the American Mathematical Society
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
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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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Additional Information

Maria Alberich-Carramiñana
Affiliation: Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028-Barcelona, Spain
Email: maria.alberich@upc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03505-6
PII: S 0002-9947(04)03505-6
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