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THE BICANONICAL MAP OF SURFACES WITH pg = 0 AND K2 [ges ] 7

Published online by Cambridge University Press:  14 June 2001

MARGARIDA MENDES LOPES  and
RITA PARDINI
MARGARIDA MENDES LOPES
Affiliation:
CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal; mmlopes@lmc.fc.ul.pt
RITA PARDINI
Affiliation:
Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy; pardini@dm.unipi.it
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Abstract

A minimal surface of general type with pg(S) = 0 satisfies 1 [les ] K2 [les ] 9, and it is known that the image of the bicanonical map φ is a surface for K2S [ges ] 2, whilst for K2S [ges ] 5, the bicanonical map is always a morphism. In this paper it is shown that φ is birational if K2S = 9, and that the degree of φ is at most 2 if K2S = 7 or K2S = 8.

By presenting two examples of surfaces S with K2S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K2S = 8 is, to our knowledge, a new example of a surface of general type with pg = 0.

The degree of φ is also calculated for two other known surfaces of general type with pg = 0 and K2S = 8. In both cases, the bicanonical map turns out to be birational.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 33 , Issue 3 , May 2001 , pp. 265 - 274
Copyright
© The London Mathematical Society 2001

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