On the Waldschmidt constant of square-free principal Borel ideals

TL;DR

This paper investigates the Waldschmidt constant of square-free principal Borel ideals, providing bounds and exact values, and demonstrates the realization of any rational number greater than or equal to one as such a constant.

Contribution

It establishes bounds and exact values for the Waldschmidt constant of these ideals and shows all rationals ≥ 1 can be realized as such constants.

Findings

Bounds for Waldschmidt constants in terms of monomial support

Exact values for specific cases

Any rational ≥ 1 can be realized as a Waldschmidt constant

Abstract

Fix a square-free monomial $m \in S = \mathbb{K}[x_1,\ldots,x_n]$. The square-free principal Borel ideal generated by $m$, denoted ${\rm sfBorel}(m)$, is the ideal generated by all the square-free monomials that can be obtained via Borel moves from the monomial $m$. We give upper and lower bounds for the Waldschmidt constant of ${\rm sfBorel}(m)$ in terms of the support of $m$, and in some cases, exact values. For any rational $\frac{a}{b} \geq 1$, we show that there exists a square-free principal Borel ideal with Waldschmidt constant equal to $\frac{a}{b}$.