m at h . A G ] 1 3 N ov 2 00 1 THE BICANONICAL MAP OF SURFACES WITH p g = 0 AND K 2 ≥ 7 , II

We study the minimal complex surfaces of general type with pg = 0 and K 2 = 7 or 8 whose bicanonical map is not birational. We show that if S is such a surface, then the bicanonical map has degree 2 (see [MP1]) and there is a fibration f : S → P 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 S = 8, then f is an isotrivial fibration with 6 double fibres, and if K S = 7, then f has 5 double fibres and it has precisely one fibre with reducible support, consisting of two components. 2000 Mathematics Classification: 14J29.