On the geography of line arrangements

TL;DR

This paper explores the combinatorial and geometric properties of line arrangements through Chern numbers, revealing new distribution results and connections to surface geography, while posing open problems in the field.

Contribution

It introduces new results on Chern slope distribution and links between accumulation points of Chern slopes and linear H-constants, advancing the understanding of line arrangement geography.

Findings

New results on the distribution of Chern slopes

Connection between accumulation points of Chern slopes and H-constants

Presentation of open problems in the geography of line arrangements

Abstract

This is a short note on various results about the combinatorial properties of line arrangements in terms of the Chern numbers of the corresponding log surfaces. This resembles the study of the geography of surfaces of general type. We prove some new results about the distribution of Chern slopes, we prove a connection between their accumulation points and the accumulation points of linear $H$-constants on the plane, and we present two open problems in relation to geography over $\mathbb Q$ and over $\mathbb C$.