Abstract
The canonical degree of a curve C on a surface X is \(K_X\cdot C\). Our main result, Theorem 1.1, 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 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 bounded negativity problem.
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Ciliberto, C., Roulleau, X. On finiteness of curves with high canonical degree on a surface. Geom Dedicata 183, 33–42 (2016). https://doi.org/10.1007/s10711-016-0142-8
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DOI: https://doi.org/10.1007/s10711-016-0142-8