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

Let $X$, $D$ be a smooth projective surface and simple normal crossing divisor on respectively. Suppose $\kappa (X, K\_X + D)\ge 0$, let $C$ an irreducible curve $X$ whose support is not contained in $\alpha$ rational number $\[ 0, 1 ]$. Following Miyaoka, we define orbibundle $\mathcal{E}\alpha$ as suitable free subsheaf of log differentials Galois cover $X$. Making use prove Bogomolov–Miyaoka–Yau inequality for the couple $(X, D+\alpha C)$. moreover that $K\_X+D$ big nef $(K\_X+D)^2$ greater than $e\_{X\setminus D}$, namely topological Euler open $X\setminus D$. As consequence inequality, by varying $\alpha$, deduce bound $(K\_X+D)\cdot C$ explicit function invariants: $(K\_X+D)^2$, D}$ $e\_{C \setminus , normalization minus points set-theoretic counterimage $D$. We finally such surfaces curves, with $- e\_{C\setminus bounded, form bounded family, particular there are only finite curves D}\le 0$.