On finiteness of curves with high canonical degree on a surface

TL;DR

This paper proves finiteness results for curves with high canonical degree and negative self-intersection on surfaces, extending to non-general type surfaces, with applications to bounded negativity and related conjectures.

Contribution

It establishes finiteness of certain curves on surfaces of general type and beyond, advancing understanding of the bounded negativity problem and related conjectures.

Findings

Finiteness of curves with large canonical degree and negative self-intersection on surfaces of general type.

Extension of finiteness results to non-general type surfaces and non-negative curves.

Application to finiteness of negative curves on blow-ups of the projective plane at many points.

Abstract

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 $n\geq 10$ 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.