Complex Hyperbolic Geometry, Arithmetic, and Commutative Algebra

University Of Utah, Salt Lake City UT

Investigators

Abstract

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 many discoveries, from special functions used to solve problems in physics and engineering, to advances in number theory and cryptography. The aim of this project is to study the special non-Euclidean geometry that has recently been discovered on the moduli spaces of cubic equations in three and four variables. The reason for carrying this study is not only the solution of the problems posed in the proposal, but also the expectation that this study of cubics in more variables will continue to be a point of departure for new ideas that will affect other areas of mathematics, science and engineering.