Plane Cremona maps, exceptional curves and roots
Author:
Maria AlberichCarramiñana
Journal:
Trans. Amer. Math. Soc. 357 (2005), 19011914
MSC (2000):
Primary 14J26, 14E05, 14E07
Published electronically:
December 10, 2004
MathSciNet review:
2115081
Fulltext PDF Free Access
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Abstract: We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of selfintersection 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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 E. CasasAlvero, Singularities of plane curves, London Math. Soc. Lecture Note Ser., vol. 276, Cambridge University Press, 2000. MR 1782072 (2003b:14035)
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 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)
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 P. Du Val, On the Kantor group of a set of points in a plane, Proc. London Math. Soc. 42 (1936), no. 2, 1851.
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 F. Enriques and O. Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche, N. Zanichelli, Bologna, 1915.
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 B. Harbourne, Blowingsup of and their blowingsdown, Duke Math. J. 52 (1985), 129148. MR 0791295 (86m:14026)
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 E. Looijenga, Rational surfaces with an anticanonical cycle, Ann. of Math. 114 (1981), 267322. MR 0632841 (83j:14030)
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Additional Information
Maria AlberichCarramiñana
Affiliation:
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028Barcelona, Spain
Email:
maria.alberich@upc.edu
DOI:
http://dx.doi.org/10.1090/S0002994704035056
PII:
S 00029947(04)035056
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 BFM2002012040, Generalitat de Catalunya 2000SGR00028 and EAGER, European Union contract HPRNCT200000099
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American Mathematical Society
