Chern degree functions and Prym semicanonical pencils

Abstract

Abelian varieties are projective algebraic varieties endowed with a group structure. They constitute one of the most explored objects in Algebraic Geometry throughout the last decades. On the one hand, abelian varieties are interesting on their own right, as they are varieties possessing a rich geometry; on the other hand, their study is useful to understand other algebraic varieties. This thesis investigates two problems motivated by the study of abelian varieties, which somehow reflect this dualism. The first problem under consideration is that of understanding cohomological rank functions on abelian surfaces. The cohomological rank functions associated to an object of the derived category of a polarized abelian variety were recently introduced by Jiang and Pareschi (building on previous work of Barja, Pardini and Stoppino), and have received several applications. Roughly speaking, these functions encode the (hyper)cohomological ranks of the object when twisted with the general representative of (possibly fractional) multiples of the polarization. In the case of elliptic curves, it is well known that the cohomological rank functions of a coherent sheaf can be described through its Harder-Narasimhan filtration. Nevertheless, for higher- dimensional abelian varieties only a few concrete examples of functions are known, and a general structure is far from being understood. In this thesis we extend the relation between the functions and stability to the case of abelian surfaces. Our main tool are Bridgeland stability conditions, which are a generalization of the classical notions of slope and Gieseker stability for sheaves in the context of the derived category. More precisely, attached to every object of the derived category of a smooth polarized surface (not necessarily abelian), we define Chern degree functions encoding the Harder-Narasimhan filtrations of the object with respect to certain stability conditions. These functions extend to continuous real valued functions, and their differentiability can be described in terms of stability. In the case of abelian surfaces, Chern degree functions recover the cohomological rank functions of Jiang and Pareschi, which gives a new insight into the problem and allows to understand aspects like their local polynomial expressions or their differentiability. This presentation is also useful for the computation of concrete examples. The most notable one corresponds to the ideal sheaf of one point, which leads to new results on the syzygies of polarized abelian surfaces. The second problem lies in the interplay between algebraic curves and abelian varieties, and has consequences on the geometry of cubic threefolds. More precisely, we deal with double étale covers of curves with a semicanonical pencil, and their Prym varieties. Such covers form two divisors in the moduli space of double étale covers, according to a certain parity condition. Adapting arguments of Teixidor for the divisor of curves with a semicanonical pencil, we prove the irreducibility of these two divisors and compute their classes in terms of the basic divisor classes of the rational Picard group of the moduli space. From the point of view of Prym varieties, the even divisor is formed by covers whose Prym variety has a singular theta divisor; however, the Prym map on the odd divisor remains largely unexplored. An analysis of the Prym map restricted to the two divisors of Prym semicanonical pencils is performed, and shows significant differences between them: whereas the Prym map on the even divisor is never dominant, in contradistinction to the odd divisor (in the cases of low genus).

Furthermore, the fibers of these restricted Prym maps often display a rich geometry. For instance, our analysis for the odd case of genus 5 has enumerative consequences on the geometry of lines on cubic threefolds.

La presente tesis aborda dos problemas distintos, enmarcados en el ámbito de las variedades abelianas. El primer objetivo es el de entender las cohomological rank functions, introducidas recientemente por Jiang y Pareschi para variedades abelianas polarizadas, en el caso de las superficies abelianas. El método aquí empleado se basa en las condiciones de estabilidad de Bridgeland en la categoría derivada. La existencia de estas condiciones de estabilidad para superficies proyectivas lisas cualesquiera (i.e. no necesariamente abelianas) nos permite introducir las Chern degree functions asociadas a objetos de la categoría derivada. Demostramos la continuidad de estas funciones como funciones de variable real y analizamos su derivabilidad en términos de estabilidad. En el caso particular de las superficies abelianas, las Chern degree functions recuperan las cohological rank functions de Jiang y Pareschi, lo cual determina una nueva estructura para estas funciones y permite describir muchos de sus aspectos. Además, esta presentación resulta útil para el cálculo de nuevos ejemplos. El más notable corresponde a las funciones del haz de ideales de un punto, de cuyo cálculo se deducen nuevos resultados sobre las syzygias de superficies abelianas polarizadas, mediante criterios recientes de Caucci e Ito. En la segunda parte se estudian los recubrimientos dobles no ramificados de curvas dotadas de un semicanonical pencil, y sus variedades de Prym. Tales recubirimientos constituyen dos divisores en el espacio de móduli de recubrimientos dobles no ramificados, para los cuales se demuestra su irreducibilidad y se calcula su clase en términos de las clases generadoras del grupo (racional) de Picard del espacio de móduli. Además, se realiza un estudio de la aplicación de Prym restringida a estos divisores, que muestra diferencias significativas entre ellos. Este análisis se realiza con especial detalle para los casos de género bajo, debido a la rica geometría que involucran. Por ejemplo, el estudio para uno de los divisores en género 5 presenta consecuencias enumerativas sobre ciertas rectas contenidas en los sólidos cúbicos.