Hirzebruch-type inequalities viewed as tools in combinatorics

TL;DR

This survey introduces Hirzebruch-type inequalities in algebraic topology and explores their applications in combinatorics, especially in point and line arrangements, highlighting recent results and open problems.

Contribution

It provides an algebro-topological overview of Hirzebruch-type inequalities and demonstrates their utility in solving combinatorial problems and conjectures.

Findings

Inequalities aid in combinatorial problem solving.

Applications include progress on the Weak Dirac Conjecture.

Open problems are identified for future research.

Abstract

The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.