Algebraic Surfaces: Rational points and Cox rings

研究者将从事算术几何方面的两个不同项目。第一个项目涉及研究在数域上定义的代数曲面的Brauer群的超越部分。具有有理坐标的点的存在和分布常常受到由该品种的Brauer群引起的上同调障碍的阻碍。这个项目的重点是K3和Enriques曲面，因为它们是具有非平凡超越Brauer类的最简单的曲面。基本的问题是找到表示在几何构造中出现的超越类的非分支Azumaya代数（例如，在K3曲面上稳定轴的模空间的扭曲泛束中）。Azumaya代数必须有一个足够具体的描述，以确定表面在不良还原处的分枝行为。PI将检查这些类在多大程度上解释这些表面上有理点的缺失或稀缺。第二个项目的重点是一个明确的研究光滑，射影有理曲面的普遍torsor。在数域上，利用泛量证明了某些上同调障碍足以解释大代数变异类上局部变全局原理的所有失效。当有限生成时，Cox环给出了通用环的显式表示。这个项目的目的是对具有有限生成的Cox环的光滑有理表面进行分类，并给出这些环的显式表示。算术几何是数论和代数几何的交叉学科：目的之一是研究多元多项式方程组的解，这些方程组的坐标都是有理数或整数（例如19/7或-5）。有时候，这些多项式系统只有很少的解，或者根本没有！该项目旨在了解阻止这些特殊解存在的现象，并对所研究的多项式方程系统进行限制。这些限制是由系统的几何形状自然产生的。对这些解决方案的研究已经充分证明了与密码学和噪声信道上的信息传输的联系。本项目不涉及这些应用；相反，它处理的是它们背后的基本问题。

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.