Linkage and Cohen-Macaulayness of Blowup Algebras

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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.