Linkage and Cohen-Macaulayness of Blowup Algebras

9970344本提案分为三个部分。第一个问题是一个与寻找具有Cohen-Macaulay爆破代数的理想密切相关的联系问题。第二章分析了Rees代数的Cohen-Macaulay性与其相应分次环的Cohen-Macaulay性的比较。用来理解这种关系的方法包括深入研究关于约化数的条件的必要性，这些条件反过来又等价于理想核心的显式公式。只要满足这样的条件，PI就会描述Rees环的一组定义方程。该方案的最后部分试图找出与其相关的分次环的深度的下界的数值条件。这是一个由代数几何推动的交换代数项目。一般来说，通过研究方程组及其结构，人们可以学到更多关于几何的知识。本课题也适合向本科生介绍主流研究。

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.