Rees algebras and singularities

等奇异性理论的一般目标是为一组分析集提供拓扑平凡的准则。理想情况下，这种标准包括数字数据，如多样性，这只取决于个体成员，而不是家庭的总空间。在先前的工作中，研究者利用复数的新概念作为数值不变量，建立了任意孤立奇点族的惠特尼等奇异性的充分条件。现在他希望证明他的等奇点条件的必要性，这将导致孤立奇点情况下惠特尼等奇点的纤维智能数值表征。此外，他还打算在等奇异性理论的背景下，提出一般的多重性理论。本文提出了一种研究投影空间中有理曲线，特别是有理平面曲线的程序，该程序通过参数化有理平面曲线的形式的协同矩阵进行研究。他希望仅从协同矩阵中提取曲线奇异点的局部信息，并了解这些奇异点的全局定位。特别地，他提出在奇点的类型和协同矩阵的形状之间建立一种对应关系，并利用这种对应关系对给定度数的有理平面曲线空间进行分层。研究者计划通过研究理想里斯代数的隐式方程来继续他的工作。理解或找到这些方程是消去理论中的一个基本而困难的问题，即使对于最简单的理想来说，这个问题也是非常开放的。研究者打算把重点放在其产生器参数化射影变量的理想。为了对隐式方程的度进行限定和理解Rees代数的Castelnuovo-Mumford正则性，他希望证明Rees代数与该变量的齐次坐标环具有相同的正则性。研究者的长期目标是明确地确定参数化变化是否为有理平面曲线的定义方程。提议的研究是在交换代数领域，这是一个数学领域，其根源在于对多变量多项式方程组的定性研究。交换代数与几何有着密切的联系，这种联系在研究者关于等奇异性和有理曲线的项目中尤为突出。多项式方程组也出现在数学之外的许多应用中。研究者关于Rees代数隐式方程的项目包含了这个应用方面。特别是，在几何建模和计算机辅助设计中，寻找参数化定义曲面的隐式方程的问题被称为隐式化问题。

A general goal in equisingularity theory is to provide criteria for a family of analytic sets to be topologically trivial. Ideally, such criteria involve numerical data, like multiplicities, that only depend on the individual members rather than the total space of the family. In prior work the investigator established a sufficient condition for the Whitney equisingularity of families of arbitrary isolated singularities, using the new notion of epsilon multiplicity as numerical invariant. Now he wishes to prove the necessity of his condition for equisingularity, which would result in a fiber-wise numerical characterization of Whitney equisingularity in the case of isolated singularities. In addition, he intends to advance the general theory of epsilon multiplicity beyond the context of equisingularity theory. The investigator proposes a program to study rational curves in projective space, most notably rational plane curves, through the syzygy matrix of the forms parametrizing them. Solely from the syzygy matrix, he wishes to extract local information about the singularities of the curve and understand the global positioning of these singularities. In particular, he proposes to set up a correspondence between the types of singularities on the one hand and the shapes of the syzygy matrix on the other hand, and to use this correspondence to stratify the space of rational plane curves of a given degree. The investigator plans to continue his work on Rees algebras of ideals by studying the implicit equations of such algebras. Understanding or finding these equations is a fundamental and difficult problem in elimination theory that is wide open even for the simplest of ideals. The investigator intends to focus on ideals whose generators parametrize projective varieties. In order to bound the degrees of the implicit equations and to understand the Castelnuovo-Mumford regularity of the Rees algebra, he wishes to prove that the Rees algebra and the homogeneous coordinate ring of the variety have the same regularity. The investigator has the long-term goal to determine the defining equations explicitly if the parametrized variety is a rational plane curve.The proposed research is in the area of Commutative Algebra, a field of mathematics that has its roots in the qualitative study of systems of polynomial equations in several variables. Commutative Algebra has close ties to geometry, a connection that is prominent in the investigator's projects on equisingularity and rational curves. Systems of polynomial equations also arise in numerous applications outside of mathematics. The investigator's project on implicit equations of Rees algebras encompasses this applied aspect. In particular, the problem of finding implicit equations of surfaces defined parametrically has relevance in geometric modeling and computer-aided design, where it is known as implicitization problem.