On the exceptional locus of the birational projections of normal surface singularity into a plane

Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $\pi : X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $\pi$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms.Comment: 12 pages, 2 figure

Teaching geography for a sustainable world: a case study of a secondary school in Spain

Geography has a major responsibility in delivering education for sustainable development (ESD), especially because the geographical concepts of place and space are key dimensions for the analysis and pursuit of sustainability. This paper presents the results of a research that investigated how the teaching of geography in secondary education in Catalonia (Spain) contributes to ESD. For the development of this research it was explored what is involved in understanding and resolving issues about sustainable development and how geography teachers might best conceptualize and teach in this new domain. As a result of this theoretical reflection it has been defined a proposal or model for reorienting the geography curriculum from the basis of the ESD paradigm, which is based and structured in four groups of criteria and recommendations as follows: recommendations for defining competences and learning objectives; criteria for selecting geographical contents and themes; criteria for selecting geographical areas and for the use of scale; and finally, recommendations for choosing the most suitable teaching and learning approach

Geometry of the Kimura 3-parameter model

The Kimura 3-parameter model on a tree of n leaves is one of the most used in phylogenetics. The affine algebraic variety W associated to it is a toric variety. We study its geometry and we prove that it is isomorphic to a geometric quotient of the affine space by a finite group acting on it. As a consequence, we are able to study the singularities of W and prove that the biologically meaningful points are smooth points. Then we give an algorithm for constructing a set of minimal generators of the localized ideal at these points, for an arbitrary number of leaves n. This leads to a major improvement of phylogenetic reconstruction methods based on algebraic geometry.Comment: 26 pages with 4 figure

A geographical issue: the contribution of Citizenship Education to the building of a European citizenship. The case of the VOICEs Comenius network

Citizenship Education is currently a consolidated issue within several European curricula. It has been integrated in national educational laws in different ways: as cross-curricular education (UK, Italy), as a subject (France, Spain) or as a skill (Ireland). Despite these differences, there is a common agreement on the ethical value of Citizenship Education and on its main aim: to foster students’ sense of local, national and European citizenship. In some ways this goal has been inspired by Morin’s path to a “plural” education and a planetary citizenship (Morin, 2000). Social sciences, and in particular Geography and History, keep the function of giving tools able to show how a dialogue among the different scales is possible. Nevertheless European citizenship is undergoing a constant redefinition due to the European enlargement process, the role of Europe inside national jurisdictions and to the changes in national curricula. This evolution directly affects the guiding function conferred to school in terms of skills, aims and themes; therefore competences and methods adopted by teachers may have to be reconsidered. This essay presents the first results of the updating of the state of the art of this issue that has been carried out by the Citizenship Education Research Group of the VOICEs Comenius network (The Voice of European Teachers). The main aim of this international research group is to face the challenge of building a European citizenship by developing a comparative analysis of teachers’ practices and strategies in different local, regional and national contexts, aiming to contribute, with renewed ideas, to the debate on this promising field of research

Equisingularity classes of birational projections of normal singularities to a plane

Given a birational normal extension S of a two-dimensional local regular ring R, we describe all the equisingularity types of the complete ideals J in R whose blowing-up has some point at which the local ring is analytically isomorphic to S. The problem of classifying the germs of such normal surface singularities was already posed by Spivakovsky (Ann. of Math. 1990). This problem has two parts: discrete and continous. The continous part is to some extent equivalent to the problem of the moduli of plane curve singularities, while the main result of this paper solves completely the discrete part.Comment: 22 pages, 3 figures. To appear in Advances in Mathematic

The Rider (2018-01-15)

https://scholarworks.utrgv.edu/rider/1072/thumbnail.jp

The Rider (2018-02-12)

https://scholarworks.utrgv.edu/rider/1076/thumbnail.jp