Arcs, Valuations, and Multiplier Ideals on Algebraic Varieties

这个研究项目属于代数几何领域。代数几何的核心是对代数变量的分类的探索，代数变量是多项式方程系统的解的集合（或这个概念的各种推广）。分类大致分为两部分：两部分分类，其中几乎在任何地方看起来都很相似的品种被组织在同一个类中，以及模理论，其中研究品种的连续变形。极小模型规划是代数变量两族分类的主要工具，在模理论中有着重要的应用。基于曲面分类模型设计的最小模型程序已成为代数几何的主要趋势之一，S. Mori因其在三维变化方面的工作而获得菲尔兹奖。最近，有了突破性的进展，使这个主题非常接近于一个完整的程序在所有方面。然而，仍有许多基本问题尚未得到解答，其中一些问题构成了本研究项目的主要目标。代数变异及其奇异性的局部研究是这一研究的一个自然组成部分。该项目有两个主要目标。第一个目标涉及1988年Shokurov的一个猜想，该猜想预测奇异点的某个局部不变量，称为最小对数差异，是半连续变化的。这一性质与涉及最小模型规划的一个关于翻转终止的猜想密切相关，并涉及到一系列关于弧空间几何及其与估值空间和Berkovich解析空间的联系的具体问题。第二个目标涉及代数变量上的乘子理想。在奇异变量上有各种乘数理想的概念，PI将探索一种统一的方法，将所有理论结合起来，并解释它们之间的联系。项目的这一部分包含了各种各样的应用，从关于法线表面对偶化束扭转的全局生成的reider型定理，到在特征0和正特征中定义的某些奇点概念之间的比较。该项目还包括教育活动和其他更广泛的影响。这些活动包括参与外展计划，各种形式的传播，包括编写一本关于代数变量的两族几何的书，以及组织从本科生暑期学校到国际会议等不同层次的活动。

This research project is in the field of algebraic geometry. At the core of algebraic geometry is the quest for classification of algebraic varieties, which are sets of solutions of systems of polynomial equations (or various generalizations of this concept). Classification roughly splits into two parts: birational classification, in which varieties that look alike almost everywhere are organized in the same class, and moduli theory, where continuous deformations of varieties are studied. The minimal model program is the major tool for the birational classification of algebraic varieties, and it has important applications to the theory of moduli. Designed over the model of classification of surfaces, the minimal model program has become one of the major trends in algebraic geometry, earning S. Mori the Fields Medal for work on three dimensional varieties. Recently, there has been groundbreaking progress which brings the subject very close to a complete program in all dimensions. There are however many fundamental questions that still remain unanswered, and some of these constitute the main objectives of this research project. The local study of algebraic varieties and their singularities is a natural component of this study.The project has two main objectives. The first objective concerns a conjecture of Shokurov from 1988 which predicts that a certain local invariant of singularities, known as the minimal log discrepancy, varies semicontinuously. This property is closely connected to a conjecture on termination of flips which has implications to the minimal model program, and relates to a series of specific questions concerning the geometry of spaces of arcs and their connections to valuation spaces and Berkovich analytic spaces. The second objective deals with multiplier ideals on algebraic varieties. There are various notions of multiplier ideals on singular varieties, and the PI will explore a unifying approach which incorporates all theories and explains their connections. This part of the project contains various applications, from a Reider-type theorem on the global generation of twists of the dualizing sheaf of a normal surface, to the comparisons between certain notions of singularities defined in characteristic zero and in positive characteristics. The project also includes educational activities and other broader impacts. These include the participation in outreach programs, various forms of dissemination including the writing of a book on the birational geometry of algebraic varieties, and the organization of several activities at various levels, ranging from undergraduate summer schools to international conferences.