Symbolic Powers, Configurations of Linear Spaces, and Applications

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

This research project is in the area of commutative algebra, with connections to algebraic geometry and computational algebra. Commutative algebra has applications in a range of areas, from statistics to game theory, from robotics to string theory. The main theme of this project is the study of configurations of linear subspaces, such as finite collections of lines in the plane. These problems are classically motivated by algebraic geometry and have received renewed interest in the last fifteen years. Despite a rapidly growing body of work, there is still much progress needed to understand the subtle behavior of these configurations. This research program aims to take advantage of the combinatorial structure inherently present in this context. The project will also study potential applications of this work to coding theory. The common thread for the investigations in this project concerns the asymptotic properties of symbolic powers. One goal of the project is the determination of certain invariants that measure these asymptotic properties (resurgence, Waldschmidt constants). Another goal is to characterize families of ideals that display extremal behavior with respect to the containment between ordinary and symbolic powers. Among the tools to be employed are methods involving the study of Rees algebras, minimal free resolutions, and local cohomology for powers of ideals. Another line of inquiry will consider the symbolic powers for singular loci of line arrangements, or more generally hyperplane arrangements, with special emphasis on reflection arrangements because of their additional structure. Despite their undoubted theoretical significance, not much is known about the practical applications of symbolic powers. The investigator and collaborators plan to start an investigation on the implications of this recent progress on symbolic powers from the point of view of applied algebraic geometry. Some computational tools in the form of scripts for the computer algebra system Macaulay will be developed to aid with the inquiry.