Algebraic Surfaces: Rational points and Cox rings

代数曲面：有理点和 Cox 环

• 批准号: 1103659
• 负责人: Anthony Varilly-Alvarado
• 金额: $ 10.11万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1103659&HistoricalAwards=false
• 关键词: Algebraic; Surfaces; Rational; points; Cox

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.

调查员将在算术几何两个不同的项目工作。第一个项目涉及研究超越部分的Brauer群的代数曲面是定义在数域。在一个簇上具有有理坐标的点的存在和分布经常受到来自该簇的Brauer群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类的最简单曲面之一。基本问题是找到表示几何构造中出现的超越类的非分歧Azumaya代数（例如，在K3曲面上的稳定层的模空间的扭泛丛中）。Azumaya代数必须有一个足够具体的描述，以确定在曲面约简不好的地方的分歧行为。PI将研究这些类在多大程度上解释了这些曲面上理性点的缺失或稀缺。第二个项目的重点是明确的研究普遍torsors的光滑，射影理性曲面。在数域上，泛扭体已被用来证明某些上同调障碍足以解释大类代数簇上局部到整体原则的所有失败。当环生成时，考克斯环给出了显式表示的通用torsor。这个项目的目的是目录类光滑的合理曲面与生成的考克斯环，并给予明确的介绍，这些rings.Arithmetic几何是一个主题，在于在交叉口的数论和代数几何：一个目标是研究系统的解决方案的多元多项式方程的坐标都是，说，有理数或整数（例如19/7，或-5）。 有时，这些多项式系统只有很少的解，或者根本没有解！ 该项目旨在了解防止这些特殊解决方案存在的现象，并对研究的多项式方程系统进行限制。这些限制自然产生于系统的几何形状。对这种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程序;相反，它涉及的基础问题，他们。