The equations of Rees algebras of equimultiple ideals of deviation one

TL;DR

This paper characterizes the defining equations of Rees algebras for a specific class of ideals, revealing a unique top-degree equation linked to the reduction number, thus extending existing algebraic results.

Contribution

It provides a detailed description of the equations of Rees algebras for equimultiple ideals of deviation one with special reduction properties.

Findings

Existence of a single top-degree equation in the minimal generating set.

Relation between the top-degree equation and the reduction number.

Recovery of several known results in the context of Rees algebra equations.

Abstract

We describe the equations of the Rees algebra R(I) of an equimultiple ideal I of deviation one, provided that I has a reduction J generated by a regular sequence and such that the initial forms of the elements of this sequence, except possibly the last one, are also a regular sequence in the associated graded ring of I. In particular, we prove that there is a single equation of top degree in a minimal generating set of the ideal of equations of R(I) and we relate this degree to the reduction number, recovering several known results in the context.