Symbolic Powers, Configurations of Linear Spaces, and Applications

这个研究项目是在交换代数的领域，与代数几何和计算代数的连接。交换代数在很多领域都有应用，从统计学到博弈论，从机器人到弦理论。这个项目的主要主题是研究线性子空间的构型，比如平面上有限的线的集合。这些问题通常是由代数几何引起的，在过去的15年里重新引起了人们的兴趣。尽管工作正在迅速增长，但要了解这些结构的细微行为，仍需要取得很大进展。本研究计划旨在利用这种情况下固有的组合结构。该项目还将研究这项工作在编码理论中的潜在应用。这个项目的研究主线是关于符号幂的渐近性质。该项目的一个目标是确定测量这些渐近性质的某些不变量（死灰复燃，Waldschmidt常数）。另一个目标是描述在普通权力和象征性权力之间的遏制方面表现出极端行为的理想家族。所使用的工具包括研究Rees代数、最小自由分辨率和理想幂的局部上同调。另一条探究线将考虑奇异轨迹线排列的符号力量，或者更一般的超平面排列，特别强调反射排列，因为它们的附加结构。尽管它们的理论意义毋庸置疑，但人们对象征性权力的实际应用知之甚少。研究者和合作者计划从应用代数几何的角度开始研究这一最新进展对符号力的影响。计算机代数系统Macaulay的一些脚本形式的计算工具将被开发来帮助探究。

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.