Commutative Algebraic Aspects of Birational Algebraic Geometry

本课题主要研究交换代数和代数几何的交界面问题。具体地说，对高维二次代数几何中的问题提出了几种交换代数方法。其中包括建立在PI关于理想余弦的先前工作的基础上，证明了Kawamata关于任何伴随着充裕丛的光滑射影簇上的任意充裕线丛的整体截面的存在的猜想。它还包括奇异环境下乘子理想理论的发展，以及对JET格式的交换代数性质的研究，特别是对于组合丰富的情况，如单项模式。本文提出的工作是更广泛的研究领域的一部分，在“纯”代数几何和交换代数，这两个领域研究几何对象(称为代数簇)，可以用多项式方程来描述。这些领域提供了将数学应用于政府和工业问题的几个重要领域的基础。例子包括基于代数簇的纠错码，以及计算机辅助设计(CAD)，它通过绘制代数簇的片断在计算机屏幕上绘制几何对象。虽然这个项目没有解决当前应用程序中的具体问题，但它确实有助于该主题的健康、蓬勃发展的基础设施，这是支持未来应用程序所必需的。此外，预计该项目的几名受训人员将在以后继续学习这一主题的更多应用方面，转到行业和政府职位(如国家安全局)。

This project focus on problems at the interface of commutative algebra andalgebraic geometry. Specifically, several commutatative algebraicapproaches to problems in higher dimensional birational algebraic geometryare proposed. These include building on the PI's prior work on ideal coresto prove a conjecture of Kawamata's on the existence of a global sectionfor any ample line bundle on a smooth projective variety which is adjointto an ample bundle. It also includes the development of the theory ofmultiplier ideals in the singular setting, and a study of the commutativealgebraic properties of jet schemes, especially for combinatorially richcases such as monomial schemes.The work proposed here is part of the broader landscape of research in"pure" algebraic geometry and commutative algebra, the fields that studythe geometric objects (called algebraic varieties) that can be describedby polynomial equations. These fields provide the underpinnings of severalimportant areas of applications of mathematics to problems in governmentand industry. Examples include error-correcting codes based on algebraicvarieties, and computer-aided design (CAD) which "draws" geometric objectson the computer screen by plotting pieces of algebraic varieties. Whilethis project does not address specific current problems in applications,it does contribute to the healthy thriving infrastructure of the subjectnecessary to support applications in the future. Furthermore, it isexpected that several of trainees in this project will continue later withmore applied aspects of the subject by moving to positions in industry andgovernment (such as the National Security Agency).