On the Waldschmidt constant of square-free principal Borel ideals

Abstract

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