Rational Points & Rational Curves on Algebraic Varieties

理性点

• 批准号: 0901777
• 负责人: Yuri Tschinkel
• 金额: $ 18万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901777&HistoricalAwards=false
• 关键词: Rational; Points; & Curves; Algebraic

This project focuses on problems at the interface of algebraic geometry and number theory, concerning connections between global geometric invariants of algebraic varieties and rational points. Among specific problems to be addressed are:effective computationsof Brauer-Manin obstructions on K3 surfaces, the study of potential density of rational and integral points on higher-dimensional varieties over number fields and function fields, and the study of their asymptotic distribution. One of the key issues is the investigation of rational curves on these varieties and their deformations.Arithmetic geometry studies integral solutions of polynomial equations in several variables with integral coefficients from a geometric point of few. One of the simplest questions is:find rectangular triangles with all three sides integers. This leads to the study of rational points on a circle of radius 1. Higher-dimensional geometric objects display a very high degree of complexity; finding and describing rational points on them poses tremendous theoretical and computational challenges. Advances in arithmetic geometry have important applications in the area of information transmission and storage, cryptography and graph theory.

该项目侧重于代数几何学和数字理论的界面上的问题，涉及代数品种的全球几何不变剂与理性点之间的联系。要解决的具体问题包括：在K3表面上的有效计算Brauer-Manin障碍物，研究对数字和功能领域的高维品种的潜在密度和积分点的潜在密度，以及对其渐近分布的研究。关键问题之一是研究这些品种及其变形的合理曲线。弧形几何研究在几个变量中具有积分系数的多项式方程的积分解决方案，这些变量来自少数几何点。最简单的问题之一是：找到所有三个侧面整数的矩形三角形。这导致对半径1圆的理性点进行研究。高维几何对象表现出非常高的复杂性。发现并描述上的理性点会带来巨大的理论和计算挑战。算术几何形状的进步在信息传输和存储，密码学和图理论领域具有重要应用。