Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces

We describe the compactifications obtained by adding slc surfaces X with ample K_X, for two connected components in the moduli space of surfaces of general type: Campedelli surfaces with \pi_1(X)=\bZ_2^3, and Burniat surfaces with K_X^2=6.Comment: Color version; black-and-white version at http://www.math.uga.edu/~valery/ap-bw.pd

On the existence of ramified abelian covers

Given a normal complete variety $Y$, distinct effective Weil divisors $D_1,... D_n$ of $Y$ and positive integers $d_1,... d_n$, we spell out the conditions for the existence of an abelian cover of $Y$ branched with order $d_i$ on $D_i$. As an application, we prove that a Galois cover of a normal complete toric variety branched on the torus-invariant divisors is itself a toric variety.Comment: Dedicated to Alberto Conte on his 70th birthday. Version 2: the assumptions in the application (Thm. 3.7) have been weakened; minor modifications of the expositio

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