On finiteness of curves with high canonical degree on a surface

The \emph{canonical degree} of a curve $C$ on a surface $X$ is $K_X\cdot C$. Our main result, is that on a surface of general type there are only finitely many curves with negative self–intersection and sufficiently large canonical degree. Our proof strongly relies on results by Miyaoka. We extend our result both to surfaces not of general type and to non–negative curves, and give applications, e.g. to finiteness of negative curves on a general blow–up of ${\mathbb{P}}^2$ at $n\ge 10$ general points (a result related to \emph{Nagata’s Conjecture}). We finally discuss a conjecture by Vojta concerning the asymptotic behaviour of the ratio between the canonical degree and the geometric genus of a curve varying on a surface. The results in this paper go in the direction of understanding the \emph{bounded negativity} problem.