Rational Points on Algebraic Varieties and Geometry of Curves and Surfaces

代数簇与曲线曲面几何上的有理点

• 批准号: 0070537
• 负责人: Brendan Hassett
• 金额: $ 6.4万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2001-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070537&HistoricalAwards=false
• 关键词: Rational; Points; Algebraic; Varieties; Geometry

This project addresses several fundamental questions inalgebraic geometry and number theory.The first part concerns the geometry of solutions toalgebraic equations.Let X be a variety defined over a number field K.Does there exist a finite extension L/K such thatX(L) is dense? One seeks criteria for when thisoccurs in terms of the geometry of X.It is still not clear what form such criteria should take. The second part concerns the geometry of plane curves.When is an abstract curve a degenerationof smooth plane curves? What geometric properties dosuch limiting plane curves have in common?The third part involves the classification of surfacesingularities. Given a family of projective normalsurfaces, when are the invariants of such a familyconstant? Specifically, how does one tell whetherthe self-intersection of the canonical divisor is constant? The last part involves finding explicitparametrizations for the curves of a fixed small genus.Algebraic geometers study the structure of solutions topolynomial equations. From ancient times, geometricfigures like circles and ellipses have fascinated architects, scientists, and artists. The most compactway to represent such a figure is to describe it as the solutions to an equation. One basic question is to classify the abstract figures that may be representedby equations of a given form. Another is to understandthe common geometric properties of the solutions toall the equations of a given form. This project seeksto answer these questions for special types of curves and surfaces.

本项目主要研究代数几何和数论中的几个基本问题.第一部分是代数方程解的几何问题.设X是定义在数域K上的一个簇，是否存在有限扩张L/K使得X（L）是稠密的？ 人们根据X的几何学寻找这种情况何时发生的标准。 第二部分是关于平面曲线的几何问题，什么时候抽象曲线是光滑平面曲线的退化曲线？ 这些极限平面曲线有什么共同的几何性质？第三部分是表面奇异性的分类。 给定一个射影正规曲面族，什么时候这个族的不变量是常数？ 具体来说，如何判断标准因子的自相交是否是常数？ 最后一部分涉及到为一个固定的小亏格的曲线寻找显式参数化。代数几何学家研究多项式方程解的结构。 自古以来，像圆形和椭圆形这样的几何图形就吸引着建筑师、科学家和艺术家。 表示这样一个图形的最简单的方法是把它描述成一个方程的解。 一个基本的问题是对可以用给定形式的方程表示的抽象图形进行分类。 另一个是理解给定形式的所有方程的解的共同几何性质。 这个项目旨在回答这些问题的特殊类型的曲线和曲面。