A Formula for the Core of Certain Strongly Stable Ideals

TL;DR

This paper derives an explicit formula for the core of a specific family of strongly stable ideals, revealing their algebraic properties and geometric significance.

Contribution

It introduces a simple explicit formula for the core of certain strongly stable ideals, expanding understanding of their algebraic and geometric properties.

Findings

Ideals satisfy an Artin-Nagata property

Fail to satisfy stronger depth conditions

Explicit core formula derived

Abstract

The core of an ideal is the intersection of all of its reductions. The core has geometric significance coming, for example, from its connection to adjoint and multiplier ideals. In general, though, the core is difficult to describe explicitly. In this paper, we investigate a particular family of strongly stable ideals. We prove that ideals in this family satisfy an Artin-Nagata property, yet fail to satisfy other, stronger standard depth conditions. We then show that there is a surprisingly simple explicit formula for the core of these ideals.