2006/03/29 by Alessandro Chiodo, Chiodo, Alessandro
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation #math.AG
paper · pdf · doi:10.48550/arxiv.math/0603689
26 pages, v2 exposition improved, introduction rewritten, a new application (Cor. 5.5.1) and a new section (\S7) added;
openalex publication_date 2006/03/29 · arxiv created 2007/04/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth curve CK over the field of fractions K. For any positive integer r prime to the residual characteristic, we consider the finite K-group scheme PicCK[r] of r-torsion line bundles on CK. We determine when there exists a finite R-group scheme, which is a model of PicCK[r] over R; in other words, we establish when the Néron model of PicCK[r] is finite. To this effect, one needs to analyse the points of the Néron model over R, which, in general, do not represent r-torsion line bundles on a semistable reduction of CK. Instead, we recast the notion of models on a stack-theoretic base: there, we find finite Néron models, which represent r-torsion line bundles on a stack-theoretic semistable reduction of CK. This allows us to quantify the lack of finiteness of the classical Néron models and finally to provide an efficient criterion for it.