Linkage and Cohen-Macaulayness of Blowup Algebras

9970344本建议书分为三部分。第一个处理的链接问题密切相关的理想科恩-麦考利爆破代数搜索。 第二章分析了Rees代数的Cohen-Macaulay性与其相应的分次环的Cohen-Macaulay性的比较。 用于理解这种关系的方法涉及到对约化数条件的必要性的深入研究，这些条件反过来又等价于理想核的显式公式。 只要满足这些条件，PI就想描述一组Rees环的定义方程。 最后一部分的建议试图找到的数值条件，暗示其相关的分次环的深度上的下界。这是一个项目，在代数几何的启发，在递归代数。 一般来说，通过研究方程组及其结构，人们可以学到更多的几何知识。 这个主题也适合介绍本科生主流研究。

9970344This proposal is divided into three parts. The first one deals with a linkage problem closely related to the search for ideals having Cohen--Macaulay blowup algebras. The second one analyzes the comparison between the Cohen-Macaulayness of the Rees algebra and the one of its associated graded ring. The methods used to understand this relationship involve a deep study of the necessity of conditions on the reduction number, which in turn are equivalent to explicit formulas for the core of an ideal. Whenever such conditions are satisfied, the PI would like to describe a set of defining equations of the Rees ring. The last part of the proposal attempts to find numerical conditions that imply lower bounds on the depth of its associated graded ring.This is a project in commutstive algebra that is motivated by algebraic geometry. In general, by studying systems of equations and their structure one can learn more about geometry. This topic is also suitable for introducing undergraduates to mainstream research.