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14j29.

Bulletin of the London Mathematical Society (1)

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THE BICANONICAL MAP OF SURFACES WITH $p_g = 0$ AND $K^2 \geqslant 7$, II
MARGARIDA MENDES LOPES, RITA PARDINI
Journal:

Bulletin of the London Mathematical Society / Volume 35 / Issue 3 / May 2003

Published online by Cambridge University Press:

12 May 2003, pp. 337-343

Print publication:

May 2003
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Minimal complex surfaces of general type with $p_g = 0$ and $K^2 = 7$ or $8$ whose bicanonical map is not birational are studied. It is shown that if $S$ is such a surface, then the bicanonical map has degree 2 (see Bulletin of the London Mathematical Society 33 (2001) 1–10) and there is a fibration $f : S \rightarrow {\bb P}^1$ such that (i) the general fibre $F$ of $f$ is a genus 3 hyperelliptic curve; (ii) the involution induced by the bicanonical map of $S$ restricts to the hyperelliptic involution of $F$.
Furthermore, if $K^2_S = 8$, then $f$ is an isotrivial fibration with six double fibres, and if $K^2_S = 7$, then $f$ has five double fibres and it has precisely one fibre with reducible support, consisting of two components.