On finiteness of curves with high canonical degree on a surface

2014/06/29 by Ciro Ciliberto, Ciliberto, Ciro, Xavier Roulleau +1
Mathematics · #14C20 #14H50 #14J #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1406.7478

openalex publication_date 2014/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The canonical degree of a curve C on a surface X is K_X⋅ 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≥ 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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