On a generalization of the Bogomolov-Miyaoka-Yau inequality and an explicit

bound for the log-canonical degree of curves on open surfaces

Pietro Sabatino
(Liceo Mamiani)

Abstract: Let X, D be respectively a smooth projective surface and a simple normal crossing divisor on X. Suppose ?(X,K_X+D) ? 0, given an irreducible curve C on X and a rational number q in [0,1], following ideas introduced by Miyaoka, we define an orbibundle E_q as a subsheaf of log differentials on a suitable Galois cover of X and prove a Bogomolov-Miyaoka-Yau inequality for this bundle. We briefly compare this construction to similar ones of e.g. Megyesi and Langer. As a consequence of this Bogovomolov-Miyaoka-Yau inequality we deduce, in the case K_X+D big and nef and (K_X+D)^2 > \chi(X \ D), a bound for (K_X+D)\cdot C by an explicit function of the invariants (K_X+D)^2, the topological Euler-Poncaré characteristic of the open surface \chi(X \ D) and \chi(\tildeC \ D), the topological Euler-Poncare' characteristic of the normalization of C minus the points mapping on D.