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

DOI:
https://doi.org/10.1090/S0002-9947-04-03505-6

Published electronically:
December 10, 2004

MathSciNet review:
2115081

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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 and , respectively) on a smooth complex projective surface which admits a birational morphism to . The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of is in the -orbit of the class of the total transform of some point blown up by if is exceptional, or in the -orbit of a simple root if is root, where is the Weyl group acting on ; 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 is a necessary and sufficient condition in order that 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:
https://doi.org/10.1090/S0002-9947-04-03505-6

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