On linear Harbourne constants

TL;DR

This paper calculates global linear Harbourne constants for up to ten lines over any field, contributing to the understanding of the Bounded Negativity Conjecture and linking algebraic geometry with combinatorics.

Contribution

It provides explicit computations of Harbourne constants over arbitrary fields for small line configurations, advancing the study of their properties and applications.

Findings

Computed Harbourne constants for configurations of up to ten lines over arbitrary fields.

Highlighted the relevance of these invariants to the Bounded Negativity Conjecture.

Connected the invariants to combinatorial aspects of line arrangements.

Abstract

In this note we compute values of global linear Harbourne constants over arbitrary fields for up to ten lines. These invariants have appeared recently in the discussions around the Bounded Negativity Conjecture. They seem to be of independent interest also from the point of view of combinatorics.