On absolute linear Harbourne constants

TL;DR

This paper investigates absolute linear Harbourne constants, providing bounds and exact values for specific configurations, thereby linking negative curve self-intersections to birational map complexity.

Contribution

It offers new bounds and exact calculations for Harbourne constants for various line configurations, extending previous results significantly.

Findings

Bounds on Harbourne constants established

Exact values computed for specific line counts

Extended previous results on Harbourne constants

Abstract

In the present note we study absolute linear Harbourne constants. These are invariants which were introduced in order to relate the lower bounds on the selfintersection of negative curves on birationally equivalent surfaces to the complexity of the birational map between them. We provide various lower and upper bounds on Harbourne constants and give their values for the number of lines $s$ of the form $p^{2r}+p^r+1$ for any prime number $p$ and also for all values of $s$ up to $31$. This extends considerably earlier results of the third author.