The icosahedral line configuration and Waldschmidt constants

Abstract

There is a highly special point configuration in P2 of 31 points, naturally arising from the geometry of the icosahedron. The 15 planes of symmetry of the icosahedron projectivize to 15 lines in P2, 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=A5×Z2, 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 P2 at the 31 points to study the line configuration. The Waldschmidt constant is a measure of how special a collection of points in P2 is. 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.