Studies on Singularities

奇点研究

• 批准号: 1202685
• 负责人: Claudia Polini
• 金额: $ 22.7万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1202685&HistoricalAwards=false
• 关键词: Studies; Singularities

The research program proposes several problems in commutative algebra. There are three long-term goals: the study of projective rational plane curves; the description of the defining equations of Rees algebras; and the investigation of cores of arbitrary ideals. More precisely, Polini proposes to investigate local and global information on the singularities of a given curve; to set up a correspondence between the types of singularities and the shapes of the syzygy matrices of the forms parametrizing them; and to stratify the space of all rational plane curves of a fixed degree according to the configuration of their singularities. Rees algebras are instrumental in multiplicity theory and intersection theory, in the study of integral closures of ideals, and in the context of blowing up a variety. Although blowing up is a basic operation, an explicit understanding of this process is still an open problem. Most notably, it is difficult to describe the defining equations of the resulting variety. Reductions play a crucial role in the study of Rees algebras, multiplicities, and Hilbert functions. To investigate all reductions at once one considers the core, defined as the intersection of these reductions. The core is related to multiplier ideals, an essential tool in birational geometry due to their importance in vanishing theorems. The core detects uniformity properties of schemes, such as the Cayley Bacharach property of finite sets of points. Although cores have been studied extensively, they remain somewhat mysterious objects that are difficult to compute in general.Commutative algebra deals with solutions of many polynomial equations in many unknowns. Fundamentally, it is the study of abstract objects called rings of polynomial functions defined on the set of solutions of systems of polynomial equations. Commutative algebra provides the tools for understanding many problems in pure and applied mathematics and, as of more recently, physics as well. In many applied problems, polynomial equations and hence commutative algebra play a crucial role. Applied areas where results from commutative algebra have been used include geometric modeling, operations research,computer science, robotics, control theory, coding theory and cryptography,to mention a few. In fact, most of the problems described in this researchproposal are not only important for commutative algebraists or algebraicgeometers but are of interest to applied mathematicians as well. For instance,the study of curve (or surface) singularities via their parametrizations have applications in geometric modeling theory. In computer aided geometric design, curves are often given parametrically and their singularities are pointswhere the shape of the graphic gets more complicated. Thus, understanding the nature of these singular points is extremely important.

该研究计划提出了交换代数中的几个问题。有三个长期目标：研究射影有理平面曲线;描述Rees代数的定义方程;以及研究任意理想的核。更确切地说，波里尼建议调查当地和全球信息的奇异性的一个给定的曲线，以建立一个对应的类型之间的奇异性和形状的syzygy矩阵的形式parametrizing他们，并分层空间的所有合理的平面曲线的一个固定的程度，根据其配置的奇异性。里斯代数是工具的多重性理论和交叉理论，在研究积分封闭的理想，并在上下文中炸毁各种。虽然爆破是一个基本的操作，但对这个过程的明确理解仍然是一个悬而未决的问题。最值得注意的是，很难描述所得到的多样性的定义方程。约化在里斯代数、重数和希尔伯特函数的研究中起着至关重要的作用。为了一次研究所有的约简，人们考虑核心，定义为这些约简的交叉点。核心是与乘数理想，一个必不可少的工具，在双有理几何由于其重要性消失定理。核心检测方案的一致性属性，例如有限点集的Cayley Bacharach属性。虽然核已经被广泛研究，但它们仍然是一些难以计算的神秘对象。交换代数处理许多未知数的许多多项式方程的解。从根本上说，它是研究抽象对象称为环的多项式函数定义的一组解决方案的多项式方程组。交换代数为理解纯数学和应用数学中的许多问题提供了工具，最近，物理学也是如此。在许多应用问题中，多项式方程和交换代数起着至关重要的作用。应用领域的结果从交换代数已被使用包括几何建模，运筹学，计算机科学，机器人，控制理论，编码理论和密码学，仅举几例。事实上，大多数的问题描述在这个researchproposal不仅是重要的交换algebraists或algebraicgeometers但感兴趣的应用数学家以及。例如，曲线（或曲面）奇异性的研究通过其参数化在几何造型理论中有应用。在计算机辅助几何设计中，曲线通常是参数化的，其奇点是图形形状变得更加复杂的点。因此，理解这些奇点的性质是极其重要的。