Curve arrangements and their H-indices

During the talk I will present how to find effective bounds for the so-called Harbourne indices for transversal curve configurations on smooth projective surfaces with non-negative Kodaira dimension. The main ingredient of our proofs is the logarithmic Bogomolov-Miyaoka-Yau inequality which allows to approach this problem without using abelian covers branched along curve configurations. I will provide some examples of curve configurations on smooth hypersurfaces in P^{^3} with extremely low H-indices. The talk is based on joint papers with Roberto Laface.