Symbolic Powers, Configurations of Linear Spaces, and Applications

这个研究项目是在交换代数领域，与代数几何和计算代数的连接。交换代数的应用范围很广，从统计学到博弈论，从机器人学到弦论。 该项目的主题是研究线性子空间的结构，例如平面上的有限直线集合。这些问题是经典的动机代数几何，并已收到新的兴趣，在过去的十五年。尽管工作量迅速增长，但要理解这些配置的微妙行为仍需要取得很大进展。该研究计划旨在利用这种背景下固有的组合结构。该项目还将研究这项工作在编码理论中的潜在应用。该项目的共同研究主题是符号幂的渐近性质。该项目的一个目标是确定某些测量这些渐近性质的不变量（复苏，Waldscherk常数）。另一个目标是表征家庭的理想，显示极端行为的遏制之间的普通和象征性的权力。其中所采用的工具是方法，涉及研究里斯代数，最小的自由决议，和当地上同调的权力的理想。另一条研究路线将考虑线排列或更一般的超平面排列的奇异轨迹的符号能力，特别强调反射排列，因为它们具有额外的结构。尽管象征性权力的理论重要性受到质疑，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调查这一最新进展对符号幂的影响。一些计算工具的形式脚本的计算机代数系统麦考利将开发，以协助调查。

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.