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

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 uniform bound on the canonical degree of Albanese defective curves on surfaces

Let S be a minimal complex surface of general type with irregularity q>=2 and let C be an irreducible curve of geometric genus g contained in S. Assume that C is "Albanese defective", i.e., that the image of C via the Albanese map does not generate the Albanese variety Alb(S); we obtain a linear upper bound in terms of K^2_S and g for the canonical degree K_SC of C. As a corollary, we obtain a bound for the canonical degree of curves with g<= q-1, thereby generalizing and sharpening the main result of [S.Y. Lu, On surfaces of general type with maximal Albanese dimension, J. Reine Angew. Math. 641 (2010), 163-175].Comment: Final version: main result generalized, title changed accordingly. To appear in Bulletin of the LM

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

A Regional Human Development Index for Portugal

In a report from 2008 the Organization for Economic Cooperation and Development came to the conclusion that Portugal is still a country very much marked by regional asymmetries and in need of better regional governance mechanisms and policies. In the face of these conclusions it becomes important to address the issue of constructing an index of regional development for Portuguese regions to better assess the evolution of the differential between regions. We propose a regional human development index for Portugal at the NUTS III level, based on the methodology of the Human Development Index (HDI) from the United Nations Development Programme (UNDP). Results show us a country that has most of the highest ranked NUTS III positioned in the coastline, although some interior NUTS III regions improve their relative positions in the ranking between 2004 and 2008. Additionally to the traditional dimensions of the HDI, we also added two dimensions, that we choose to include, given the main criticisms pointed in the literature to the HDI - governance and environment. Results show some significative differences when we add the environment dimension, but in terms of governance they don't change significantly.Human Development Index, Regional Asymmetries, Portugal.