Complex Hyperbolic Geometry, Arithmetic, and Commutative Algebra

主要研究者研究了三次曲面和三次曲面的模空间的复双曲几何。对于复数上的稳定三次曲面，D. Allcock等人先前的工作给出了一个由爱森斯坦整数上定义的格从模空间到四球商的周期映射，这是这两个空间之间的同构，并且将奇异曲面的模子空间化为球中的超平面的一个位形。研究者现在研究周期为爱森斯坦有理点的三次曲面的算术性质。对于复数上的稳定立方三倍，存在一个周期映射，从它们的模空间到十球的商，这个周期映射也是在爱森斯坦整数上定义的晶格。为了证明这个周期映射是两个空间之间的同构（经过一些爆破和爆破），研究者研究了这个周期映射的详细性质，它将奇异三倍模的子空间携带到球中的超平面构型上。研究的主要技术要点是参数化等奇异变化的各层周期图约束的微分计算。将格里菲思剩余演算推广到这种情况是相当复杂的，需要交换代数中的新技术。对三次方程的理解一直是数学的中心主题，对数学、科学和工程有着深远的影响。在16世纪，单变量三次方程的解是通过找到一个类似于众所周知的二次方程解的公式来求解的。这个公式导致了复数的引入，它现在是科学和工程的标准工具。在18和19世纪，人们认识到，任何数量的变量的二次方程都可以被理解，而且，对于固定数量的变量，所有的二次方程通过变量的变化本质上是等价的。人们还认识到，两个或多个变量的三次方程通过变量变换并不都是等价的。不同的等价类现在被称为模空间。二元三次方程的模空间是非欧几里德几何的双曲平面。这种联系导致了许多发现，从用于解决物理和工程问题的特殊函数，到数论和密码学的进步。本项目的目的是研究最近在三变量和四变量三次方程的模空间上发现的特殊的非欧几里得几何。进行这项研究的原因不仅是解决了提议中提出的问题，而且还期望对更多变量的立方的研究将继续成为影响数学，科学和工程其他领域的新思想的出发点。

The principal investigators study the complex hyperbolic geometry of the moduli spaces of cubic surfaces and cubic threefolds. For stable cubic surfaces over the complex numbers, previous work of D. Allcock and the investigators gives a period map from the moduli space to the quotient of the four-ball by a lattice defined over the Eisenstein integers, which is an isomorphism between these two spaces and which takes the subspace of moduli of singular surfaces to a configuration of hyperplanes in the ball. The investigators now study the arithmetic properties of cubic surfaces whose periods are Eisenstein rational points. For stable cubic threefolds over the complex numbers, there is a period map from their moduli space to the quotient of the ten-ball by a lattice also defined over the Eisenstein integers. The investigators study detailed properties of this period map in order to prove that this map is an isomorphism between the two spaces (after some blowing up and down) which carries the subspace of moduli of singular threefolds to a hyperplane configuration in the ball. The main technical point under study is the computation of the differential of the restriction of the period map to each stratum that parametrizes equisingular varieties. The extension of the Griffiths residue calculus to this situation turns out to be quite involved and to require new techniques in commutative algebra.The understanding of cubic equations has been a central theme in mathematics, whith deep implications for mathematics, science and engineering. In the sixteenth century cubic equations in one variable were solved by finding a formula for its solutions analogous to the well-known formula for the solutions of quadratic equations. This formula led to the introduction of complex numbers, which are now a standard tool in science and engineering. In the eighteenth and nineteenth centuries it was realized that quadratic equations in any number of variables could be understood, and that, for a fixed number of variables, all quadratic equations are essentially equivalent by a change of variables. It was also realized that cubic equations in two or more variables are not all equivalent by change of variables. The different equivalence classes are now called the moduli space. The moduli space of cubic equations in two variables is the hyperbolic plane of non-Euclidean geometry. This connection has led to manydiscoveries, from special functions used to solve problems in physics andengineering, to advances in number theory and cryptography. The aim of thisproject is to study the special non-Euclidean geometry that has recently beendiscovered on the moduli spaces of cubic equations in three and fourvariables. The reason for carrying this study is not only the solution ofthe problems posed in the proposal, but also the expectation that this studyof cubics in more variables will continue to be a point of departure for newideas that will affect other areas of mathematics, science and engineering.