745 research outputs found
A note on a theorem of Xiao Gang
In 1985 Xiao Gang proved that the bicanonical system of a complex surface
of general type with is not composed of a pencil [Bull. Soc. Math.
France, 113 (1985), 23--51]. When in the end of the 80's it was finally proven
that is base point free, whenever , the part of this
theorem concerning surfaces with became trivial.
In this note a new proof of this theorem for surfaces with is
presented.Comment: Latex, 4 page
Enriques surfaces with eight nodes
A nodal Enriques surface can have at most 8 nodes. We give an explicit
description of Enriques surfaces with 8 nodes, showing that they are quotients
of products of elliptic curves by a group isomorphic to or to
acting freely in codimension 1. We use this result to show that if is a
minimal surface of general type with such that the image of the
bicanonical map is birational to an Enriques surface then and the
bicanonical map is a morphism of degree 2.Comment: Latex 2e, 11 page
The bicanonical map of surfaces with and , II
We study the minimal complex surfaces of general type with and
or 8 whose bicanonical map is not birational. In the paper 'The
bicanonical map of surfaces with and ' we have shown that if
is such a surface, then the bicanonical map has degree 2. Here we describe
precisely such surfaces showing that there is a fibration f\colon S\to \pp^1
such that: i) the general fibre of is a genus 3 hyperelliptic curve;
ii) the involution induced by the bicanonical map of restricts to the
hyperelliptic involution of . Furthermore, if , then is an
isotrivial fibration with 6 double fibres, and if , then has 5
double fibres and it has precisely one fibre with reducible support, consisting
of two components.Comment: Latex 2e, 8 page
A survey on the bicanonical map of surfaces with and
We give an up-to-date overview of the known results on the bicanonical map of
surfaces of general type with and .Comment: LaTeX2e, 12 pages. To appear in the Proceedings of the Conference in
memory of Paolo Francia, Genova, september 200
A new family of surfaces with and
Let S be a minimal complex surface of general type with p_g=0 such that the
bicanonical map of S is not birational and let Z be the bicanonical image. In
[M.Mendes Lopes, R.Pardini, "Enriques surfaces with eight nodes", Math. Zeit.
241 (4) (2002), 673-683] it is shown that either: i) Z is a rational surface,
or ii) K^2_S=3, the bicanonical map is a degree two morphism and Z is
birational to an Enriques surface. Up to now no example of case ii) was known.
Here an explicit construction of all such surfaces is given.
Furthermore it is shown that the corresponding subset of the moduli space of
surfaces of general type is irreducible and uniruled of dimension 6.Comment: Latex, 36 page
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