The icosahedral line configuration and Waldschmidt constants

TL;DR

This paper investigates a special 31-point configuration in the projective plane derived from the icosahedron's symmetry, analyzing invariant curves to compute the Waldschmidt constant associated with these points.

Contribution

It introduces a detailed study of the icosahedral line configuration's symmetry group action and computes the Waldschmidt constant for the associated point ideal.

Findings

Identification of negative G-invariant curves on the blow-up

Explicit computation of the Waldschmidt constant for the 31 points

Insight into the geometric structure of the icosahedral configuration

Abstract

There is a highly special point configuration in $\mathbb{P}^2$ of 31 points, naturally arising from the geometry of the icosahedron. The 15 planes of symmetry of the icosahedron projectivize to 15 lines in $\mathbb{P}^2$, whose points of intersections yield the 31 points. Each point corresponds to an opposite pair of vertices, faces or edges of the icosahedron. The symmetry group of the icosahedron is $G=A_5\times \mathbb{Z}_2$, one of finitely many exceptional complex reflection groups. The action of $G$ on the icosahedron descends onto an action on the line configuration. We blow up $\mathbb{P}^2$ at the 31 points to study the line configuration. The Waldschmidt constant is a measure of how special a collection of points in $\mathbb{P}^2$. In this paper, we study negative $G$-invariant curves on this blow-up in order to compute the Waldschmidt constant of the ideal of the $31$ singularities.