Rees algebras and singularities

等价性理论的一个一般目标是给出一族分析集拓扑平凡的准则。理想情况下，这样的标准涉及数字数据，比如多重性，只取决于单个成员，而不是整个家庭的空间。在以前的工作中，研究者利用epsilon重数的新概念作为数值不变量，建立了任意孤立奇异族的Whitney均衡性的一个充分条件。现在，他希望证明他的等奇性条件的必要性，这将导致在孤立奇点的情况下，惠特尼等奇性的纤维数值表征。此外，他还打算将epsilon多重性的一般理论推向超越均衡性理论的范围。研究人员提出了一个程序来研究射影空间中的有理曲线，特别是有理平面曲线，通过将它们参数化的形式的合合矩阵。仅从合成矩阵中，他希望提取有关曲线奇点的局部信息，并了解这些奇点的全局位置。特别是，他建议建立奇点类型和合力矩阵形状之间的对应关系，并利用这种对应关系对给定次数的有理平面曲线空间进行分层。这位研究者计划通过研究理想的Rees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，这个问题也是开放的。研究者打算把重点放在生成元参数化投射簇的理想上。为了界定隐式方程的次数，了解Rees代数的Castelnuovo-Mumford正则性，他想证明Rees代数与簇的齐次坐标环具有相同的正则性。研究人员的长期目标是明确地确定定义方程是否为有理平面曲线。拟议的研究是在交换代数领域，这是一个数学领域，其根源于对多元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值性和有理曲线的项目中是突出的。多项式方程组也出现在数学之外的许多应用中。这位研究人员关于Rees代数隐式方程的项目就包含了这一应用方面。特别地，求解参数定义的曲面的隐式方程的问题在几何造型和计算机辅助设计中具有重要意义，称为隐含化问题。

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.