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The Bicanonical Map of Surfaces with p g = 0 and K 2 ⩾ 7, II
Author(s) -
Mendes Lopes Margarida,
Pardini Rita
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609302001819
Subject(s) - fibration , mathematics , hyperelliptic curve , involution (esoterism) , genus , mathematics subject classification , pure mathematics , combinatorics , algebra over a field , botany , homotopy , politics , political science , law , biology
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 → 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, ifK S 2 = 8 , then f is an isotrivial fibration with six double fibres, and ifK 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.