Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constants

TL;DR

This paper investigates the geometry of symmetric line configurations in the projective plane, focusing on negative curves on blowups, and computes resurgence and Waldschmidt constants for associated ideals, revealing rare containment failures.

Contribution

It introduces a detailed study of invariant negative curves on blowups of the plane at symmetric configurations, and precisely computes resurgence and Waldschmidt constants for these ideals.

Findings

Exact resurgence for the Wiman configuration

Exact Waldschmidt constant for the Wiman configuration

Estimates for resurgence and Waldschmidt constants for the Klein configuration

Abstract

The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity at least three. In this paper we study the surface X obtained by blowing up the projective plane in the singular points of one of these line configurations. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X. The homogeneous ideal I of the collection of points in the configuration is an example of an ideal where the symbolic cube of the ideal is not contained in the square of the ideal; ideals with this property are seemingly quite rare. The resurgence and asymptotic resurgence are invariants which were introduced to measure such failures of containment. We use our knowledge of negative curves on X to compute the resurgence of I exactly. We also compute the asymptotic resurgence and Waldschmidt constant exactly in the case of the Wiman configuration of lines, and provide estimates on both for the Klein configuration.