Studies on Singularities

本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；Rees代数定义方程的描述；以及对任意理想核心的研究。更准确地说，Polini建议研究给定曲线奇点的局部和全局信息；建立奇异点的类型与参数化奇异点形式的合矩阵的形状之间的对应关系；并根据其奇点的构形对所有定次有理平面曲线的空间进行分层。Rees代数在多重性理论和交点理论、理想的积分闭包的研究以及在膨胀变化的背景下都是有用的。虽然爆破是一项基本操作，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变化的定义方程。约简在Rees代数、复数和希尔伯特函数的研究中起着至关重要的作用。为了一次调查所有的削减，我们考虑核心，它被定义为这些削减的交集。核心是关于乘数理想的，由于它们在消失定理中的重要性，乘数理想是二分几何中必不可少的工具。核心检测方案的均匀性，如有限点集的Cayley Bacharach性质。尽管对地核进行了广泛的研究，但总的来说，它们仍然是难以计算的神秘物体。交换代数处理许多多项式方程在许多未知数中的解。从根本上说，它是研究被称为多项式函数环的抽象对象，它被定义在多项式方程组的解的集合上。交换代数为理解纯数学和应用数学以及最近的物理学中的许多问题提供了工具。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。交换代数结果的应用领域包括几何建模、运筹学、计算机科学、机器人、控制理论、编码理论和密码学等。事实上，本研究计划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，曲线（或曲面）奇点的参数化研究在几何建模理论中有应用。在计算机辅助几何设计中，曲线通常是参数化的，其奇异点是图形形状变得更加复杂的点。因此，理解这些奇异点的本质是极其重要的。

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.