Hirzebruch-type inequalities viewed as tools in combinatorics

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 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 Weak Dirac's Conjecture. We advertise also some open problems and questions.

A Sylvester-Gallai result for concurrent lines in the complex plane

“…Beyond the Fermat configurations, there are two known exceptional configurations in C 2 : the Klein configuration with 21 points and the Wiman configuration with 45 points. For more details on these configurations, see [1,6,11].…”