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

### Pietro Sabatino

Università di Roma Tor Vergata, Italy

## Abstract

Let $X$, $D$ be a smooth projective surface and a simple normal crossing divisor on $X$, respectively. Suppose $κ(X,K_{X}+D)≥0$, 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 $E_{α}$ as a suitable free subsheaf of log differentials on a Galois cover of $X$. Making use of $E_{α}$ we prove a Bogomolov–Miyaoka–Yau inequality for the couple $(X,D+αC)$. Suppose moreover that $K_{X}+D$ is big and nef and $(K_{X}+D)_{2}$ is greater than $e_{X∖D}$, 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 $(K_{X}+D)⋅C$ by an explicit function of the invariants: $(K_{X}+D)_{2}$, $e_{X∖D}$ and $e_{C∖D}$ , 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 $−e_{C∖D}$ bounded, form a bounded family, in particular there are only a finite number of curves $C$ on $X$ such that $−e_{C∖D}≤0$.

## Cite this article

Pietro Sabatino, An Explicit Bound for the Log-Canonical Degree of Curves on Open Surfaces. Publ. Res. Inst. Math. Sci. 58 (2022), no. 4, pp. 817–853

DOI 10.4171/PRIMS/58-4-6