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

Abstract

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