THE BICANONICAL MAP OF SURFACES WITH $p_g = 0$ AND $K^2 \geqslant 7$, II

Abstract

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.