Quantitative Néron theory for torsion bundles
Abstract
Description
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 Pic_{CK}[r] of r-torsion line bundles on CK. We determine when there exists a finite R-group scheme, which is a model of Pic_{CK}[r] over R; in other words, we establish when the Néron model of Pic_{CK}[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.
26 pages, v2 exposition improved, introduction rewritten, a new application (Cor. 5.5.1) and a new section (\S7) added;
26 pages, v2 exposition improved, introduction rewritten, a new application (Cor. 5.5.1) and a new section (\S7) added;