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 $-e_{C\setminus D}\le 0$.