Rational points on surfaces

January 8–15, 2013 — Concepción

Schedule

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 by Martin Bright, Ronald van Luijk, and the author (Warwick, 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.

Topics include:

• The classical Segre–Manin Theorem on unirationality of del Pezzo surfaces (degree ≥ 2)
• Recent results with Cecília Salgado and Tony Várilly‑Alvarado
• Cox rings of rational surfaces
• References:
  • Joint work with Tony Várilly‑Alvarado and Mauricio Velasco
  • Work of Michela Artebani and Antonio Laface
  • Joint work with Antonio Laface

Second lecture: Rational points on K3 surfaces

Among surfaces of vanishing Kodaira dimension, K3 surfaces occupy a special role; many basic questions are known and many are still wide open.

Topics include:

• Methods for computing Picard groups of surfaces
• Discussion of K3 surfaces appearing in a paper by Ronald van Luijk
• Joint work with Michela Artebani and Antonio Laface on a K3 surface arising from a problem studied by Büchi

Third lecture: Rational points on surfaces of general type

Surfaces of general type are often mysterious with respect to rational points, especially when there is no connection to abelian varieties. For example, no simply connected surface of general type over a number field is known to have a non‑empty and explicitly determined set of rational points.

Topics include:

• Examples from classical number‑theoretic problems
• Moduli spaces of abelian surfaces (see Hulek–Sankaran’s notes on Siegel modular threefolds)
• Connection to the surface of cuboids
  • Ronald van Luijk’s undergraduate thesis
  • More recent joint work with Michael Stoll