3,218 research outputs found
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 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 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
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.
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