Rees algebras and singularities

里斯代数和奇点

• 批准号: 1205002
• 负责人: Bernd Ulrich
• 金额: $ 25.68万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205002&HistoricalAwards=false
• 关键词: Rees; algebras; singularities

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.

等奇性理论的总体目标是为解析集族提供拓扑平凡的标准。理想情况下，此类标准涉及数字数据，例如多重性，仅取决于个体成员而不是家庭的总空间。在之前的工作中，研究者使用 epsilon 多重性的新概念作为数值不变量，为任意孤立奇点族的惠特尼等奇异性建立了充分条件。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对惠特尼等奇性进行纤维数值表征。此外，他打算将 epsilon 多重性的一般理论推进到等奇异性理论的范围之外。研究人员提出了一个程序，通过参数化形式的 syzygy 矩阵来研究射影空间中的有理曲线，尤其是有理平面曲线。他希望仅从 syzygy 矩阵中提取有关曲线奇点的局部信息并了解这些奇点的全局定位。特别是，他提出一方面在奇点的类型与另一方面的syzygy矩阵的形状之间建立对应关系，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。研究人员计划通过研究此类代数的隐式方程来继续他对理想里斯代数的研究。理解或找到这些方程是消除理论中的一个基本且困难的问题，即使对于最简单的理想也是如此。研究人员打算重点关注其生成器参数化射影簇的理想。为了限制隐式方程的次数并理解里斯代数的Castelnuovo-Mumford正则性，他希望证明里斯代数和簇的齐次坐标环具有相同的正则性。如果参数化变量是有理平面曲线，研究者的长期目标是明确确定定义方程。拟议的研究属于交换代数领域，这是一个数学领域，其根源在于系统的定性研究多个变量的多项式方程。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。研究者关于里斯代数隐式方程的项目涵盖了这个应用方面。特别是，寻找参数化定义的曲面的隐式方程的问题与几何建模和计算机辅助设计相关，被称为隐式化问题。