A note on a theorem of Xiao Gang

In 1985 Xiao Gang proved that the bicanonical system of a complex surface $S$ of general type with $p_2(S)>2$ 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 $| 2K_S|$ is base point free, whenever $p_g\geq 1$, the part of this theorem concerning surfaces with $p_g\geq 1$ became trivial. In this note a new proof of this theorem for surfaces with $p_g=0$ 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 $\Z_2^2$ or to $\Z_2^3$ acting freely in codimension 1. We use this result to show that if $S$ is a minimal surface of general type with $p_g=0$ such that the image of the bicanonical map is birational to an Enriques surface then $K^2_S=3$ and the bicanonical map is a morphism of degree 2.Comment: Latex 2e, 11 page

The bicanonical map of surfaces with pg=0p_g=0 and K2≥7K^2\ge 7, II

We study the minimal complex surfaces of general type with $p_g=0$ and $K^2=7$ or 8 whose bicanonical map is not birational. In the paper 'The bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$' we have shown that if $S$ 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 $F$ 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 6 double fibres, and if $K^2_S=7$, then $f$ 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 pg=0p_g=0 and K2≥2K^2\ge 2

We give an up-to-date overview of the known results on the bicanonical map of surfaces of general type with $p_g=0$ and $K^2\ge 2$.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 pg=0p_g=0 and K2=3K^2=3

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