2019/01/08 by Pietro Sabatino, Sabatino, Pietro
Arts and Humanities · Mathematics · Social Sciences · #14C17 #Algebraic Geometry (math.AG) #FOS: Mathematics #French Historical and Cultural Studies #Historical Studies and Socio-cultural Analysis #Primary 14J29 #Secondary 14J60 #Vietnamese History and Culture Studies #math.AG #msc:14C17 #msc:14J29 #msc:14J60
paper · pdf · doi:10.48550/arxiv.1901.02541
24 pages, to appear in Publ. RIMS Kyoto Univ
openalex publication_date 2019/01/08 · arxiv created 2021/06/03 · arxiv updated 2021/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X, D be a smooth projective surface and a simple normal crossing divisor on X, respectively. Suppose κ(X, KX + D)≥ 0, let C be an irreducible curve on X whose support is not contained in D and α a rational number in [ 0, 1 ]. Following Miyaoka, we define an orbibundle Eα as a suitable free subsheaf of log differentials on a Galois cover of X. Making use of Eα we prove a Bogomolov-Miyaoka-Yau inequality for the couple (X, D+αC). Suppose moreover that KX+D is big and nef and (KX+D)2 is greater than eX∖ D, namely the topological Euler number of the open surface X∖ D. As a consequence of the inequality, by varying α, we deduce a bound for (KX+D)⋅ C) by an explicit function of the invariants: (KX+D)2, eX∖ D and eC ∖ 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 - eC∖ D bounded form a bounded family, in particular there are only a finite number of curves C on X such that - eC∖ D≤ 0.