# An explicit bound for the log-canonical degree of curves on open surfaces

## TL;DR

This paper establishes an explicit bound for the log-canonical degree of curves on open surfaces using a Bogomolov-Miyaoka-Yau inequality for pairs, leading to finiteness results for certain bounded curves.

## Contribution

It introduces a new inequality for pairs involving a surface, a divisor, and a curve, providing explicit bounds and finiteness results for curves with bounded topological Euler number.

## Key findings

- Derived an explicit bound for (K_X+D)·C in terms of invariants
- Proved a Bogomolov-Miyaoka-Yau inequality for log pairs
- Established finiteness of curves with bounded topological Euler number

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