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Rationality and Irrationality in Families of Varieties

品种族中的理性与非理性

基本信息

批准号：
1701659
负责人：
Brendan Hassett
金额：
$ 17.94万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701659&HistoricalAwards=false
关键词：
Rationality Irrationality Families Varieties

项目摘要

When can we write down all the solutions of a polynomial equation? We seek equations that can be parametrized with rational functions. These are used in mapmaking (stereographic projection), computer graphics, and modeling problems. Indeed, parametrizations are often the most efficient way to render geometric objects as screen images. Mathematicians have developed a rich theory for deciding when such parametrizations are possible. This project will advance this theory.An algebraic variety is rational if it can be obtained from projective space by a sequence of algebraic modifications. Given a family of smooth complex projective varieties, it is difficult to say which members are rational or irrational, or even to formulate qualitative results about the locus of rational members. The PI will use the technique of decomposition of the diagonal, and related tools from deformation theory, Hodge theory and classical geometry, to shed light on this question. The PI and his collaborators have recently shown that rationality is not a deformation invariant property: There are families of smooth complex projective varieties with both rational and irrational members. Despite examples, like cubic fourfolds, that have been extensively studied, there are no cases where the loci of rational and irrational members have been precisely described. The PI will refine and make effective the decomposition of the diagonal technique to clarify which members of a family are irrational - can they be expressed in Hodge-theoretic terms? At the same time, the PI will advance constructive techniques to exhibit rational and unirational parametrizations with prescribed properties.

什么时候我们可以写出一个多项式方程的所有解？我们寻找可以用有理函数参数化的方程。它们用于地图制作（球极投影），计算机图形学和建模问题。实际上，参数化通常是将几何对象渲染为屏幕图像的最有效方法。数学家们已经发展出了丰富的理论来决定何时可以进行这种参数化。一个代数簇是有理的，如果它可以从射影空间通过一系列代数修改得到。给定一个光滑复射影簇族，很难说哪些成员是理性的或非理性的，甚至很难说出关于理性成员轨迹的定性结果。PI将使用对角线分解技术，以及变形理论，霍奇理论和经典几何的相关工具，来阐明这个问题。PI和他的合作者最近证明了合理性不是变形不变的性质：存在光滑复射影簇的家族，其中既有合理的成员，也有不合理的成员。尽管像三次四折这样的例子已经被广泛研究过，但还没有一个例子能精确地描述出理性和非理性成员的轨迹。PI将细化并有效地分解对角线技术，以澄清一个家庭的哪些成员是无理数-他们可以用霍奇理论的术语表示吗？与此同时，PI将推进建设性技术，以展示具有指定属性的有理和单有理参数化。