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

Abstract

A \(\Bbbk \)-configuration of type \((d_1,\ldots ,d_s)\), where \(1\leqslant d_1< \cdots < d_s \) are integers, is a 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 \(\Bbbk \)-configurations in \({\mathbb P}^2\) are determined by the type \((d_1,\ldots ,d_s)\). However the Waldschmidt constant of a \(\Bbbk \)-configuration in \({\mathbb P}^2\) of the same type may vary. In this paper, we find that the Waldschmidt constant of a \(\Bbbk \)-configuration in \({\mathbb P}^2\) of type \((d_1,\ldots ,d_s)\) with \(d_1\ge s\ge 1\) is s. Then we deal with the Waldschmidt constants of standard \(\Bbbk \)-configurations in \({\mathbb P}^2\) of type (a), (a, b), and (a, b, c) with \(a\ge 1\). In particular, we prove that the Waldschmidt constant of a standard \(\Bbbk \)-configuration in \({\mathbb P}^2\) of type (1, b, c) with \(c\ge 2b+2\) does not depend on c.