Algebraic Surfaces: Rational points and Cox rings

William Marsh Rice University, Houston TX

Investigators

Abstract

The investigator will work on two different projects in arithmetic geometry. The first project involves studying the transcendental part of the Brauer group of an algebraic surface that is defined over a number field. The existence and distribution of points with rational coordinates on a variety is often hampered by cohomological obstructions arising from the Brauer group of the variety. The focus of this project is on K3 and Enriques surfaces, as these are among the simplest kind of surfaces that have nontrivial transcendental Brauer classes. The basic problem is to find unramified Azumaya algebras representing transcendental classes that arise in geometric constructions (e.g., in the twisted universal bundles of moduli spaces of stable sheaves on K3 surfaces). The Azumaya algebras must have a description that is concrete enough to determine the ramification behavior at places of bad reduction for the surface. The PI will examine the extent to which these classes explain the absence or scarcity of rational points on such surfaces. The second project focuses on an explicit study of universal torsors of smooth, projective rational surfaces. Over number fields, universal torsors have been used to prove that certain cohomological obstructions suffice to explain all failures of local-to-global principles on large classes of algebraic varieties. When finitely generated, Cox rings give rise to explicit presentations of universal torsors. The aim of this project is to catalogue classes of smooth rational surfaces with finitely generated Cox rings and give explicit presentations for these rings. Arithmetic geometry is a subject that lies at the crossroads of number theory and algebraic geometry: one aim is to study the solutions of a system of multivariate polynomial equations whose coordinates are all, say, rational numbers or integers (e.g. 19/7, or -5). Sometimes, these systems of polynomials have very few such solutions, or none at all! This project seeks to understand the phenomena that prevent the existence of these special solutions, with restrictions on the systems of polynomials equations studied. These restrictions arise naturally from the geometry of the systems. The study of such solutions has amply documented connections to cryptography and the transmission of information on noisy channels. This project does not address these applications; rather, it deals with foundational questions that underly them.