Complex Hyperbolic Geometry, Arithmetic, and Commutative Algebra

主要研究者研究三次曲面和三次三重模空间的复双曲几何。 对于复数域上稳定的三次曲面，D. Allcock和研究者给出了一个从模空间到四球商的周期映射，这个映射是在Eisenstein整数上定义的格，它是这两个空间之间的同构，并且把奇异曲面的模的子空间带到球中的超平面的配置。 研究人员现在研究的算术性质的三次曲面的周期是爱森斯坦有理点。 对于复数上稳定的三次三重数，有一个周期映射，从它们的模空间到十球的商，通过一个也定义在爱森斯坦整数上的格。 研究人员研究了这个周期映射的详细性质，以证明这个映射是两个空间之间的同构（经过一些上下爆破），它将奇异三重模的子空间携带到球中的超平面配置。 研究的主要技术点是计算周期图对每个层的限制的微分，参数化等奇异变化。 将Griffiths剩余演算推广到这种情形是相当复杂的，需要交换代数中的新技术。对三次方程的理解一直是数学中的一个中心问题，对数学、科学和工程都有着深刻的意义。 在十六世纪三次方程在一个变量解决了找到一个公式，其解决方案类似于众所周知的公式的解决方案的二次方程。 这个公式导致了复数的引入，复数现在是科学和工程中的标准工具。 在18世纪和19世纪，人们认识到，可以理解任何数量变量的二次方程，并且对于固定数量的变量，所有的二次方程通过变量的变化基本上是等价的。 人们还认识到，在两个或两个以上的变量的三次方程是不等价的变量的变化。 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元和四元三次方程的模空间上发现的特殊的非欧几何。 进行这项研究的原因不仅是解决提案中提出的问题，而且还期望更多变量的三次函数的研究将继续成为影响数学，科学和工程其他领域的新的出发点。

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.