The Waldschmidt constant of a standard $\mathbb{k}$-configuration in ${\mathbb{P}}^2$

Abstract

A $\mathbb{k}$-configuration of type $(d_1,\dots ,d_s)$ is a specific set of points in ${\mathbb{P}}^2$ that has a number of algebraic and geometric properties. For example, the graded Betti numbers and Hilbert functions of all $\mathbb{k}$-configurations in ${\mathbb{P}}^2$ are determined by the type $(d_1,\dots ,d_s)$. However the Waldschmidt constant of a $\mathbb{k}$-configuration in ${\mathbb{P}}^2$ of the same type may vary. In this paper, we find that the Waldschmidt constant of a $\mathbb{k}$-configuration in ${\mathbb{P}}^2$ of type $(d_1,\dots ,d_s)$ with $d_1\ge s\ge 1$ is $s$. We also find the Waldschmidt constant of a standard $\mathbb{k}$-configuration in ${\mathbb{P}}^2$ of type $(a,b,c)$ with $a\ge 1$ except the type $(2,3,5)$. In particular, we prove that the Waldschmidt constant of a standard $\mathbb{k}$-configuration in ${\mathbb{P}}^2$ of type $(1,b,c)$ with $c\ge 2b+2$ does not depend on $c$.