Rees algebras and singularities

等奇异性理论的一个一般目标是为一族解析集提供拓扑平凡的准则。理想情况下，这样的标准涉及数字数据，如多重性，只取决于单个成员，而不是家庭的总空间。在以前的工作中，研究者建立了一个充分条件的惠特尼equisingularity家庭的任意孤立奇点，使用新的概念，作为数值不变量的多重性。现在，他希望证明他的条件的必要性为equisingularity，这将导致一个纤维明智的数值表征惠特尼equisingularity的情况下，孤立的奇点。此外，他打算在等奇异性理论的背景之外推进epsilon多重性的一般理论。研究人员提出了一个程序来研究合理的曲线在射影空间，最显着的合理的平面曲线，通过syzygy矩阵的形式参数化他们。仅仅从syzygy矩阵，他希望提取局部信息的奇异性的曲线和了解全球定位这些奇点。特别是，他建议建立一个对应的类型之间的奇异性，一方面和形状的syzygy矩阵的另一方面，并利用这种对应分层空间的合理平面曲线的一个给定的程度。调查计划继续他的工作里斯代数的理想通过研究隐式方程等代数。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也是开放的。调查人员打算把重点放在理想的发电机parametrize投射品种。为了约束度的隐式方程和理解的Castelnuovo-Mumford正则性的里斯代数，他希望证明，里斯代数和齐次坐标环的品种有相同的规律性。研究者的长期目标是明确地确定定义方程，如果参数化的品种是一个合理的平面curve.The拟议的研究是在交换代数，数学领域，有其根源的定性研究系统的多项式方程在几个变量的领域。交换代数与几何学有着密切的联系，这种联系在研究者关于等奇异性和有理曲线的项目中非常突出。多项式方程组也出现在数学之外的许多应用中。调查员的项目隐含方程的里斯代数包括这方面的应用。特别是，寻找参数化定义的曲面的隐式方程的问题在几何建模和计算机辅助设计中具有相关性，其中它被称为隐式化问题。

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.