Waldschmidt constants of symmetric sets of points in $\mathbb{P}^3$

TL;DR

This paper computes Waldschmidt constants for symmetric point configurations in three-dimensional projective space, extending known results from plane configurations and addressing new challenges in higher dimensions.

Contribution

It provides exact Waldschmidt constants for configurations derived from specific root systems in $ ext{P}^3$, advancing the understanding of these invariants in higher dimensions.

Findings

Waldschmidt constants computed for $D_4$, $B_4$, $F_4$, and $H_4$ configurations in $ ext{P}^3$

New methods developed for higher-dimensional point configurations

Enhanced understanding of invariants related to complex reflection groups

Abstract

Configurations of points defined by complex reflection groups have attracted a lot of attention recently in several directions of research, e.g., the containment problem between ordinary and symbolic powers of ideals, in the theory of unexpected hypersurfaces, in the study of sets of points whose general projections are complete intersections and in the ideas revolving around the Bounded Negativity Conjecture. In order to understand these configurations better several attempts have been undertaken to compute their various invariants. In this paper we focus on their Waldschmidt constants. In the case of plane configurations of points determined by reflection groups most (but not all) Waldschmidt constants are known. Here we pass to configurations in $\mathbb{P}^3$ where new ideas are required as the identification between divisors and curves is no longer available. In particular, we precisely compute the Waldschmidt constant for configurations of points in $\mathbb{P}^3$ coming from the $D_4,B_4,F_4$, and $H_4$ root systems.