The Waldschmidt constant of square-free principal Borel ideals

Abstract

Fix a square-free monomial $m = x_{i_1} \cdots x_{i_s}$ in $S = \K[x_1, \dots, x_n]$, where $\K$ is a field. The ideal generated by all square-free monomials that are Borel moves of $m$ is referred to as the \emph{square-free principal Borel ideal}, denoted by $\mathrm{sfBorel}(m)$. While the Waldschmidt constant of $\mathrm{sfBorel}(m)$ has been partially studied in the literature, we consider the remaining cases and provide bounds for the Waldschmidt constant of $\mathrm{sfBorel}(m)$. In some cases, we explicitly compute its exact value. Furthermore, we study this invariant for square-free Borel and square-free lex-segment ideals, establishing both bounds and exact values for special cases.