Some line and conic arrangements and their Waldschmidt constants

TL;DR

This paper investigates the Waldschmidt constants of certain point configurations in the projective plane, providing exact values for sets with many collinear points and characterizing configurations with bounded Waldschmidt constants.

Contribution

It offers a complete geometric characterization of point sets with specific Waldschmidt constants and describes configurations with many points on a conic under certain bounds.

Findings

Waldschmidt constant values for sets with most points on a line

Complete geometric characterization of such point sets

Descriptions of configurations with many points on a conic

Abstract

We study the Waldschmidt constant of some configurations in the projective plane. In the first part, we show that the Waldschmidt constant of a set $\mathbb{X}$ of $n$ points where at least $n-3$ points among them lie on a line is either equal to $1, \frac{2n-3}{n-1}, 2, \frac{16}{7}, \frac{7}{3}, \frac{17}{7},$ or $\frac{5}{2}$. Together with the Hilbert polynomials, this gives a complete geometric characterization for $\mathbb{X}$. Next, we study some specific configurations whose Waldschmidt constants are bounded from above by $\frac{5}{2}$. Under this condition, we describe all configurations of $n$ points with $n-1$ points among them lying on an irreducible conic, and we also study some specific configurations of $9$ points.