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。它有三个长期目标：研究射影有理平面曲线；描述Rees代数的定义方程；研究任意理想的核。更确切地说，波利尼建议调查给定曲线奇点的局部和全局信息；建立奇点类型和将奇点参数化的形式的合力矩阵的形状之间的对应关系；并根据奇点的配置对所有固定次数的有理平面曲线的空间进行分层。Rees代数在重数理论和交集理论中，在研究理想的积分闭包，以及在爆破簇的背景下都是有用的。虽然炸毁是一项基本操作，但对这一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，很难描述由此产生的各种定义方程。约化在Rees代数、重数和Hilbert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化，我们考虑核心，定义为这些约化的交集。核心是与乘子理想有关的，这是二次几何中的一个基本工具，因为它们在消失定理中的重要性。核检测方案的一致性性质，例如有限点集的Cayley Bacharach性质。虽然核已经得到了广泛的研究，但它们仍然是某种神秘的物体，一般很难计算。交换代数处理许多未知数中的许多多项式的解。从根本上说，它是研究定义在多项式方程组解的集合上的抽象对象，称为多项式函数环。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，最近也为理解物理学提供了工具。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。应用交换代数的结果的领域包括几何建模、运筹学、计算机科学、机器人学、控制理论、编码理论和密码学，仅举几例。事实上，这个研究方案中描述的大多数问题不仅对交换代数学家或代数几何学家很重要，而且对应用数学家也很感兴趣。例如，通过曲线(或曲面)的参数化来研究曲线(或曲面)奇点在几何造型理论中有应用。在计算机辅助几何设计中，曲线往往是以参数形式给出的，其奇点是图形形状变得更加复杂的点。因此，了解这些奇点的性质是极其重要的。