manuscripta mathematica

, Volume 129, Issue 3, pp 337–368

Quantitative Néron theory for torsion bundles

Article

Abstract

Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth, geometrically connected, and projective 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 \({{\rm Pic}_{C_K}[r]}\) of r-torsion line bundles on CK. We determine when there exists a finite R-group scheme, which is a model of \({{\rm Pic}_{C_K}[r]}\) over R; in other words, we establish when the Néron model of \({{\rm Pic}_{C_K}[r]}\) is finite. The obvious idea would be to study the points of the Néron model over R, but in general these 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.

Mathematics Subject Classification (2000)

14h40 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institut Fourier, U.M.R. CNRS 5582, U.F.R. de MathématiquesUniversité de Grenoble 1Saint Martin d’HèresFrance

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