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

TL;DR

This paper determines the Waldschmidt constant for certain $K$-configurations in $P^2$, revealing how it varies with configuration type and providing explicit values for many cases.

Contribution

It explicitly computes the Waldschmidt constant for standard $K$-configurations in $P^2$ of various types, extending understanding of their algebraic properties.

Findings

Waldschmidt constant equals s for type $(d_1, o,d_s)$ with $d_1 o s$

Waldschmidt constant for type $(a,b,c)$ with $a o 1$, except $(2,3,5)$

Constant does not depend on c when type is $(1,b,c)$ with $c o 2b+2$

Abstract

A $\Bbbk$-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 $\Bbbk$-configurations in $\mathbb P^2$ are determined by the type $(d_1,\dots,d_s)$. 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 $(d_1,\dots,d_s)$ with $d_1\ge s\ge 1$ is $s$. We also find the Waldschmidt constant of a standard $\Bbbk$-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 $\Bbbk$-configuration in $\mathbb P^2$ of type $(1,b,c)$ with $c\ge 2b+2$ does not depend on $c$.