Emmy Noether: una contribución extraordinaria y generosa al establecimiento de la Geometría Algebraica

En la charla se mostrará la importante contribución de la investigación desarrollada por Emmy Noether sobre la teoría de ideales a los fundamentos de la Geometría Algebraica. Su alumno Van der Waerden tuvo un papel clave en la consolidación de esta rama de la Geometría y veremos algunos resultados e ideas de Emmy Noether que dieron lugar posteriormente a líneas de investigación y avan-ces significativos en las etapas más contemporáneas de la relación del Álgebra Conmutativa y la Geometría.La Factoria FM

Inflectional loci of scrolls

Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let \L=\mathcal O_{\mathbb P^N}(1)|_X denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of \L are locally free. The inflectional loci of $X$ can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.Comment: 9 pages, improved version. Accepted in Mathematische Zeitschrif

Osculatory behavior and second dual varieties of del Pezzo surfaces

Let X be a projective manifold, embedded via a line bundle L. For any point x∈X the k-th projective tangent bundle is defined as Tk,x=P(H0(L/mk+1x)). There is a natural map jk from the global sections H0(L) to Tk,x, assigning to each section its k-th jet jk,x(s) at x. After choosing local parameters x1,…,xn, around the point x, jk,x(s) is given by the t-tuple of coefficients of the Taylor expansion of s around x, truncated at the k-th degree. The k-th osculatory space at k is given by Ok,x=P(im(jk)). When the map jk is onto at x, the line bundle is said to be k-jet ample at x, i.e. the osculatory space Ok,x=Tk,x has maximal rank. It is natural to try to give a characterization of the osculatory spaces to a given variety, polarized by a given line bundle. R. Piene and X. S. Dai [in Enumerative geometry (Sitges, 1987), 215–224, Lecture Notes in Math., 1436, Springer, Berlin, 1990; MR1068967 (91k:14040)] studied the osculatory behavior of balanced rational normal scrolls. D. Perkinson [Michigan Math. J. 48 (2000), 483–515; MR1786502 (2001h:14066)] gave a combinatorial characterization of the rank of the osculatory space for nonsingular toric varieties. In the paper under review the authors analyze the second osculatory behavior of del Pezzo surfaces polarized by the anticanonical line bundle. Del Pezzo surfaces are constructed by blowing up P2 in d points in general positions, for d=0,…,8. For d=0 the second and the third osculatory space are of maximal rank at every point. The authors prove that for d=1,2,3,4 the second osculatory space is generically of maximal rank. The locus where the rank drops to 5 and 4 is given by the union of the exceptional divisors and the proper transforms of lines joining two points blown up. For d=5 it is proven that the rank is never maximal; it is always 5 unless the point belongs to the intersection of an exceptional divisor and a proper transform of a line joining two points blown up. For d=6 the rank is 4 unless the point belongs to the intersection of three coplanar lines of a smooth cubic surface in P3. The second osculatory spaces parametrize hyperplane sections which are singular at x, with multiplicity ≥3. Using the results on the second osculatory spaces, the authors are able to give a detailed description (degree, lower-dimensional components) of the second dual variet

Nuevos retos en educación matemática: soluciones creativas

El minicurso tiene dos partes: una dedicada a la Resolución de Problemas y otra consistente en una sesión sobre “Nuevos retos en Educación Matemática” en la cual se exponen las líneas de trabajo recientes de la Comisión de Educación de la Real Sociedad Matemática Española (RSME), sus análisis y sus propuestas. Primera parte: ponencia sobre Resolución de Problemas con ejemplos “no convencionales” cuyas soluciones fomentan la creatividad dentro de la Matemática Elemental.Segunda parte: Coordinada por Luis J. Rodríguez-Muñiz y Raquel Mallavibarrena. Se exponen reflexiones y propuestas sobre los aspectos prioritarios de los últimos años dentro de las tareas de la Comisión de Educación de la RSME