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

2003; Wiley; Volume: 35; Issue: 03 Linguagem: Inglês

10.1112/s0024609302001819

1469-2120

Margarida Mendes Lopes, Rita Pardini,

Geometric and Algebraic Topology

Minimal complex surfaces of general type with pg = 0 and K2 = 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 → P1 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 2 = 8 , then f is an isotrivial fibration with six double fibres, and if K S 2 = 7 , then f has five double fibres and it has precisely one fibre with reducible support, consisting of two components. 2000 Mathematics Subject Classification 14J29.