The orbifold Langer-Miyaoka-Yau inequality and Hirzebruch-type inequalities

Abstract

To Professor Kamil Rusek, on the occasion of his 70th birthday. Abstract Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality [8, Theorem 0.1] we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.

The 21 Reducible Polars of Klein’s Quartic

“…Not much is known about H-indices of conic-line arrangements. In a very recent paper [27], the first author proved the following result. .…”

“…([27, Theorem 2.1]) Let CL = {L 1 , ..., L l , C 1 , ..., C k } be an arrangement of l lines and k conics such that t r = 0 for r > 2(l+2k) 3…”

Extremal properties of line arrangements in the complex projective plane

“…In this section we are going to construct an abelian cover branched along conic-line arrangements having only ordinary singularities in order to obtain a Hirzebruch-type inequality. It is worth mentioning in this place that the first author has obtained an inequality in this spirit in [18] with use of an orbifold Miyaoka-Yau inequality. However, it turned out that this result is not easily applicable in the context of possible applications, for instance towards the local negativity phenomenon, or the theory of log-Chern slopes.…”

Hirzebruch-type inequalities viewed as tools in combinatorics

“…It is natural to ask whether one can find an improvement of the above inequality, for instance in order to have the so-called quadratic right-hand side. It turns out that this can be achieved with help of Langer's ideas around his version of the orbifold Miyaoka-Yau inequality [31,Theorem 2.2].…”