On the Waldschmidt constant of square-free principal Borel ideals

Fix a square-free monomial $m\in S=\mathbb{K}[x_1,\dots ,x_n]$ . The square-free principal Borel ideal generated by $m$ , denoted $\mathrm{sfBorel}(m)$ , 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 $\mathrm{sfBorel}(m)$ in terms of the support of $m$ , and in some cases, exact values. For any rational $\frac{a}{b}\ge 1$ , we show that there exists a square-free principal Borel ideal with Waldschmidt constant equal to $\frac{a}{b}$ .