Seshadri constants and special configurations of points in the projective plane

TL;DR

This paper investigates properties of special plane curves and point configurations to derive bounds and computations for multi-point Seshadri constants on the complex projective plane, especially at singular points of line arrangements.

Contribution

It provides new lemmas for plane curves and bounds for Seshadri constants when points are not very general, focusing on line arrangement singularities.

Findings

Lower bounds for multi-point Seshadri constants at specific point configurations

Identification of extremal lines in line arrangements that compute Seshadri constants

Extension of Ein-Lazarsfeld-Xu-type lemmas to new settings

Abstract

In the present note, we focus on certain properties of special curves that might be used in the theory of multi-point Seshadri constants for ample line bundles on the complex projective plane. In particular, we provide three Ein-Lazarsfeld-Xu-type lemmas for plane curves and a lower bound on the multi-point Seshadri constant of $\mathcal{O}_{\mathbb{P}^{2}}(1)$ under the assumption that the chosen points are not very general. In the second part, we focus on certain arrangements of points in the plane which are given by line arrangements. We show that in some cases the multi-point Seshadri constants of $\mathcal{O}_{\mathbb{P}^{2}}(1)$ centered at singular loci of line arrangements are computed by lines from the arrangement having some extremal properties.