2007/11/05 by Giancarlo Urzúa, Giancarlo Urzua, Urzua, Giancarlo
Mathematics · #14J29 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.NT #msc:14J29
paper · pdf · doi:10.48550/arxiv.0711.0765
Revised version which includes a new record for Chern ratios of simply connected smooth projective surfaces of general type
openalex publication_date 2007/11/05 · arxiv created 2008/06/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a strong relation between Chern and log Chern invariants of algebraic surfaces. For a given arrangement of curves, we find nonsingular projective surfaces with Chern ratio arbitrarily close to the log Chern ratio of the log surface defined by the arrangement. Our method is based on sequences of random p-th root covers, which exploit a certain large scale behavior of Dedekind sums and lengths of continued fractions. We show that randomness is necessary for our asymptotic result, providing another instance of "randomness implies optimal". As an application over the complex numbers, we construct nonsingular simply connected projective surfaces of general type with large Chern ratio. In particular, we improve the Persson-Peters-Xiao record for Chern ratios of such surfaces.