Equigenerated ideals of analytic deviation one

TL;DR

This paper investigates conditions under which the special fiber of an equigenerated homogeneous ideal with analytic deviation one is Cohen--Macaulay, focusing on graded rings and introducing new technical conditions.

Contribution

It introduces the concepts of analytical tightness and analytical adjustment to study Cohen--Macaulayness of the special fiber in graded rings with analytic deviation one.

Findings

Established conditions for Cohen--Macaulayness of the special fiber

Applied results specifically to ideals with analytic deviation one in dimension three

Provided new insights into the structure of equigenerated ideals in graded settings

Abstract

The overall goal is to approach the Cohen--Macaulay property of the special fiber $\mathcal{F}(I)$ of an equigenerated homogeneous ideal $I$ in a standard graded ring over an infinite field. When the ground ring is assumed to be local, the subject has been extensively looked at. Here, with a focus on the graded situation, one introduces two technical conditions, called respectively, {\em analytical tightness} and {\em analytical adjustment}, in order to approach the Cohen--Macaulayness of $\mathcal{F}(I)$. A degree of success is obtained in the case where $I$ in addition has analytic deviation one, a situation looked at by several authors, being essentially the only interesting one in dimension three. Naturally, the paper has some applications in this case.