Artin-Nagata properties, minimal multiplicities, and depth of fiber cones

TL;DR

This paper explores the relationship between minimal multiplicities of ideals, the Cohen-Macaulay property of fiber cones, and introduces Goto-minimal j-multiplicity, providing bounds on reduction numbers under certain conditions.

Contribution

It introduces Goto-minimal j-multiplicity for ideals of maximal analytic spread and studies its implications for fiber cone Cohen-Macaulayness and reduction number bounds.

Findings

Introduces Goto-minimal j-multiplicity for ideals.

Establishes conditions linking minimal multiplicities and Cohen-Macaulay fiber cones.

Provides bounds on the reduction number for specific ideals.

Abstract

Minimal values of multiplicities of ideals have a strong relation with the depth of blowup algebras. In this paper, we introduce the notion of Goto-minimal $j$-multiplicity for ideals of maximal analytic spread. In a Cohen-Macaulay ring, inspired by the work of S. Goto, A. Jayanthan, T. Puthenpurakal, and J. Verma, we study the interplay among this new notion, the notion of minimal $j$-multiplicity introduced by C. Polini and Y. Xie, and the Cohen-Macaulayness of the fiber cone of ideals satisfying certain residual assumptions. We are also able to provide a bound on the reduction number of ideals of Goto-minimal $j$-multiplicity having either Cohen-Macaulay associated graded algebra, or linear decay in the depth of their powers.