An Explicit Bound for the Log-Canonical Degree of Curves on Open Surfaces

Let X, D be a smooth projective surface and a simple normal crossing divisor on X, respectively. Suppose κ(X, KX + D) ≥ 0, let C be an irreducible curve on X whose support is not contained in D and α a rational number in [0, 1]. Following Miyaoka, we define an orbibundle Eα as a suitable free subsheaf of log differentials on a Galois cover of X. Making use of Eα we prove a Bogomolov–Miyaoka–Yau inequality for the couple (X, D+αC). Suppose moreover that KX +D is big and nef and (KX +D)2 is greater than eX\D, namely the topological Euler number of the open surface X \ D. As a consequence of the inequality, by varying α, we deduce a bound for (KX +D)·C by an explicit function of the invariants: (KX +D)2, eX\D and eC\D, namely the topological Euler number of the normalization of C minus the points in the set-theoretic counterimage of D. We finally deduce that on such surfaces curves, with −eC\D bounded, form a bounded family, in particular there are only a finite number of curves C on X such that −eC\D ≤ 0.