Rationality and Irrationality in Families of Varieties

我们什么时候才能写下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。它们用于地图绘制(赤平投影)、计算机图形学和建模问题。事实上，参数化通常是将几何对象呈现为屏幕图像的最有效方式。数学家们已经发展出丰富的理论来决定这种参数化何时可能实现。如果一个代数簇可以通过一系列的代数修改从射影空间中得到，则它是有理的。给定一族光滑复杂的射影变种，很难判断哪些成员是有理的或无理的，甚至很难得出关于有理成员轨迹的定性结果。PI将使用对角线的分解技术，以及形变理论、霍奇理论和经典几何的相关工具来阐明这个问题。PI和他的合作者最近表明，理性不是变形不变的性质：存在既有理性成员又有无理成员的光滑复射族。尽管像三次四折这样的例子已经得到了广泛的研究，但还没有精确地描述有理和无理成员的轨迹的情况。PI将改进并有效地分解对角线技术，以澄清一个家族中的哪些成员是无理的--它们可以用霍奇理论的术语来表达吗？同时，PI将推进构造性技术，以展示具有指定性质的有理和单调的参数化。

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.