Rational Points on Surfaces


Rational points on surfaces

January 8-15, 2013 — Concepción

8 January, 2013 15:00 - 17:00 FM-208 Rational points on rational surfaces
11 January, 2013 15:00 - 17:00 FM-208 Rational points on K3 surfaces
15 January, 2013 15:00 - 17:00 Sala Postg. Rational points on surfaces of general type

Abstract: The goal of the minicourse is to present different methods that have been used to analyze the set of rational points on smooth projective surfaces over fields in general, typically number fields or finite fields. The three lectures are independent of one another and they all start with a history and background section, followed by recent developments. Since it is easy to analyze the change in the set of rational points of a smooth projective surface under a birational transformation, it is common to reduce the study to minimal models (over the ground field), and to treat the various cases of the classification of surfaces separately. As a reference to the subject see the lecture notes for a minicourse Martin Bright, Ronald van Luijk and myself organized in Warwick in 2008.

First lecture: Rational points on rational surfaces.

By a result of Iskovskikh, a rational surface over a field is birational to either a del Pezzo surface or a conic bundle over a conic. Heuristically, over number fields, if a rational surface has points, they tend to be dense and often can be completely parameterized. I will talk about the classical Segre-Manin Theorem on unirationality of del Pezzo surfaces of degree at least two and report on recent results obtained in collaboration with Cecília Salgado and Tony Várilly-Alvarado. I will also talk about Cox rings of rational surfaces and mention joint work with Tony Várilly-Alvarado and Mauricio Velasco, work of Michela Artebani and Antonio Laface, as well as joint work with Antonio Laface.

Second lecture: Rational points on K3 surfaces

Among the surfaces of vanishing Kodaira dimension, K3 surfaces play a very special role; many basic questions are known and many are still unknown. First, I will discuss the general method for computing Picard groups of surfaces with an emphasis on K3 surfaces appearing in a paper of R. van Luijk. Then, I will talk about joint work with Michela Artebani and Antonio Laface on a specific K3 surface arising from a problem studied by Büchi.

Third lecture: Rational points on surfaces of general type

Surfaces of general type are very mysterious from the point of view of their rational points, especially when there is no direct connection with abelian varieties; for instance, there is no simply connected surface of general type over a number field having non-empty and completely explicit set of rational points. I will give some examples of surfaces arising from classical number theoretic questions, and will then focus on moduli spaces of abelian surfaces (see Klaus Hulek and Greg Sankaran's notes on Siegel modular threefolds for an introduction) and mention their relationship to the surface of cuboids (see Ronald van Luijk's undergraduate thesis and a more recent joint work with Michael Stoll).