Curve configurations in the projective plane and their characteristic numbers

TL;DR

This paper investigates characteristic numbers and Chern slopes of curve configurations in the projective plane, revealing extremal configurations and bounds that relate to key conjectures in algebraic geometry.

Contribution

It introduces bounds on characteristic numbers for certain curve configurations and explores their implications for the bounded negativity conjecture and surface geography.

Findings

Extremal line configurations share the same asymptotic invariants.

Characteristic numbers for certain smooth curve configurations are bounded by 8/3.

Connections between bounded negativity and surface geography are illuminated.

Abstract

In this paper we study the concept of characteristic numbers and Chern slopes in the context of curve configurations in the real and complex projective plane. We show that some extremal line configurations inherit the same asymptotic invariants, namely asymptotic Chern slopes and asymptotic Harbourne constants which sheds some light on relations between the bounded negativity conjecture and the geography problem for surfaces of general type. We discuss some properties of Kummer extensions, especially in the context of ball-quotients. Moreover, we prove that for a certain class of smooth curve configurations in the projective plane their characterstic numbers are bounded by $8/3$.