Prym varieties and the canonical map of surfaces of general type

Let X be a smooth complex surface of general type such that the image of the canonical map $\phi$ of X is a surface $\Sigma$ and that $\phi$ has degree $\delta\geq 2$. Let $\epsilon\colon S\to \Sigma$ be a desingularization of $\Sigma$ and assume that the geometric genus of S is not zero. Beauville has proved that in this case S is of general type and $\epsilon$ is the canonical map of S. Beauville has also constructed the only infinite series of examples $\phi:X\to \Sigma$ with the above properties that was known up to now. Starting from his construction, we define a {\em good generating pair}, namely a pair $(h:V\to W, L)$ where h is a finite morphism of surfaces and L is a nef and big line bundle of W satisfying certain assumptions. We show that by applying a construction analogous to Beauville's to a good generating pair one obtains an infinite series of surfaces of general type whose canonical map is 2-to-1 onto a canonically embedded surface. In this way we are able to construct more infinite series of such surfaces. In addition, we show that good generating pairs have bounded invariants and that there exist essentially only 2 examples with $\dim |L|>1$. The key fact that we exploit for obtaining these results is that the Albanese variety P of V is a Prym variety and that the fibre of the Prym map over P has positive dimension.Comment: 40 pages, LaTeX 2.0

Perfilru: un programa para el análisis de la rugosidad de perfiles microtopográficos mediante el estudio de la geometría fractal

[Resumen] Presentamos un programa de ordenador para el estudio de la rugosidad de perfiles y su variación con la escala. El análisis se basa en la obtención de la dimensión fractal del perfil para intervalos de escala seleccionables, calculada mediante métodos de regresión lineal, lo que permite su análisis estadístico. Este método ha sido puesto a punto para estudiar los efectos meteorizantes causados por ciclos de helada experimentales sobre muestras de roca arcillosa.[Abstract] A computer program for the analysis of profile roughness and its variation with scale is presented. The analysis is based on the calculation of the fractal dimension of the profile for selectable ranges of scale, performed through lineal regression method, that enables further statistical analysis. This program has been set up in order to study the weathering effects of experimental freezing cicles on mudrock samples

On the Hopf Problem and a Conjecture of Liu-Maxim-Wang

We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu, Maxim, and Wang in [LMW21]. In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.Comment: This new version includes some changes following the feedback of J\'anos Koll\'ar. Added acknowledgments and references updated. 12 pages, no figure

Effects of electronic correlation on X-Ray absorption and dichroic spectra at L2,3_{2,3} edge

We present a new theoretical approach to describe X-Ray absorption and Magnetic Circular Dichroism spectra in the presence of e-e correlation. Our approach provides an unified picture to include correlations in both charged and neutral excitations, namely in direct / inversion photoemission where electrons are removed/added, and photo absorption where electrons are promoted from core levels to empty states. We apply this approach to the prototypical case of L$_{2,3}$ edge of 3$d$ transition metals and we show that the inclusion of many body effects in the core level excitations is essential to reproduce, together with satellite structures in core level photoemission, the observed asymmetric line shapes in X-ray absorption and dichroic spectra.Comment: 3 figures, 5 pages, submitted to Phys. Rev.