The Waldschmidt constant of a standard \(\Bbbk \)-configuration in \({\mathbb P}^2\)

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