Program
Mini-course 1: Pawel Borowka
(Jagiellonian University, Poland)
Prym varieties and Prym maps
The notion of the Jacobian of a curve and the Torelli theorem provide a connection between complex curves and abelian varieties. In particular, the construction of coverings of curves leads to non-simple abelian varieties and hence complementary abelian subvarieties. The Prym variety of a covering is the complementary subvariety of the Jacobian of the bottom curve in the Jacobian of the top curve. Then the Prym map associates to a covering its Prym variety. Some recent results show that in many cases the Prym map is injective.
During the mini-course we plan to:
1. introduce coverings of curves with a focus to hyperelliptic coverings;
2. introduce complementary abelian subvarieties and Prym maps;
3. show results concerning some cases of a Prym map and state open problems.
Mini-course 2: Sorin Dumitrescu
(Université Côte d'Azur, France)
Holomorphic SL(2,C)-connections over Riemann surfaces
This mini-course is about second order differential equations on Riemann surfaces and the associated geometric structures, namely the complex projective structures, which played a crucial role in the uniformization of Riemann surfaces.
In this mini-course we plan to introduce and address the following:
1. Elliptic equations and their role in the uniformization of elliptic curves; Riccati equations; complex projective structures.
2. The Schwarzien operator and various parametrizations of the space of complex projective structures.
3. Prescribing real or cocompact monodromy for SL(2,C)-connections over the rank two trivial bundle over Riemann surfaces. Application: the existence of holomorphic curves of genus g ≥ 2 in quotients of SL(2,C).
Arnaud Beauville
(Université Côte d'Azur, France)
Integrable systems in algebraic geometry
Completely integrable systems, a classic subject in mechanics, have been recently highlighted in algebraic geometry. I will explain how they appear and discuss briefly some examples (hyperkähler manifolds, Hitchin fibration). Then I'll explain my recent result (with Etesse, Höring, Liu, Voisin) on intersection of two quadrics and completely integrable system.
Paola Comparin
(Universidad de La Frontera, Chile)
Quasismooth varieties
A variety Y in an ambient toric variety X is called quasismooth if the singular points of Y are in the irrelevant locus of X. This is equivalent to smoothness in projective spaces, but not in general. In this talk I will present results for quasismoothness of hypersurfaces in toric varieties and its characterization using combinatorial properties of the Newton polytope of the hypersurface. I will also talk about the relations of quasismoothness with being a Calabi-Yau variety and the possible generalizations to higher codimension. This is a joint work with M. Artebani and R. Guilbot.
Natalia García-Fritz
(Pontificia Universidad Católica, Chile)
On the geometric case of Campana's conjecture
In arithmetic, Campana points are a far reaching generalization of the notion of powerful number. In recent years, they have been the center of much attention in connection with conjectures of Campana, Manin and Vojta. Their geometric counterpart are curves in a variety endowed with an effective divisor, that meet the divisor with high multiplicity at each intersection point.
Campana's conjecture in this geometric context predicts a precise bound for the degree of such curves in terms of the genus, analogous to Demailly's notion of algebraic hyperbolicity. In this talk I will explain a recent approach to the geometric Campana conjecture based on algebraic differential equations.
Milagros Izquierdo
(Linköping University, Sweden)
On the Connectivity of Branch Loci of Spaces of Curves
Since the 19th century the theory of Riemann surfaces has a central place in mathematics putting together complex analysis, algebraic and hyperbolic geometry, group theory and combinatorial methods.
Since Riemann, Klein and Poincaré among others, we know that a compact Riemann surface is a complex curve, and also the quotient of the hyperbolic plane by a Fuchsian group.
In this talk we study the connectivity of the moduli spaces of Riemann surfaces (i.e. in spaces of Fuchsian groups). Spaces of Fuchsian groups are orbifolds where the singular locus is formed by Riemann surfaces with automorphisms: the branch loci. With a few exceptions the branch loci is disconnected and consists of several connected components.
This talk is an introductory survey of the different methods, concepts and topics playing together in the theory of Riemann surfaces.
Marcos Jardim
(Universidade Estadual de Campinas, Brasil)
The Bourbaki number of a plane curve
The study of free plane algebraic curves has a long history in algebraic geometry and commutative algebra, going back to the work of Saito in the 1980s. After briefly revising this theory, I will introduce the Bourbaki number of a plane curve, and invariant that measures how far from being free a given curve is. I will also discuss three 0-dimensional schemes naturally associated with a curve and a syzygy for its jacobian, explaining how they are related to each other.
Juan Carlos Naranjo
(Universitat de Barcelona, Spain)
Sophie Germain type coverings of curves of genus 2
We consider unramified cyclic coverings of odd degree d = 2k + 1 of curves of genus 2. By a result of Lange and Ortega, it is known that the corresponding Prym map P_d has degree 10 for d = 7, and Albano and Pirola proved that the generic fibers of P_3 and P_5 are positive dimensional. Moreover, Agostini proved that P_d is generically finite for d ≥ 7. In this talk I will report on a proof of the generic injectivity for P_d assuming that d and k are prime numbers. The primes k such that also 2k + 1 is a prime number are known as Sophie Germain primes. Our method is based on the study of the isogeny type of the Prym variety and the computation of the theta dual variety of some distinguished curves.
This is a joint work with A. Ortega and I. Spelta.
Pietro Speziali
(Universidade Estadual de Campinas, Brasil)
On Riemann surfaces completely determined by the order of their automorphism group
Abstract: click here