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Quantitative Néron theory for torsion bundles

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Abstract

Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth, geometrically connected, and projective curve C K 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 C K . 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 C K . 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 C K . This allows us to quantify the lack of finiteness of the classical Néron models and finally to provide an efficient criterion for it.

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Authors and Affiliations

  1. Institut Fourier, U.M.R. CNRS 5582, U.F.R. de Mathématiques, Université de Grenoble 1, BP 74, 38402, Saint Martin d’Hères, France

    Alessandro Chiodo

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  1. Alessandro Chiodo
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Correspondence to Alessandro Chiodo.

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A. Chiodo was financially supported by the Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme, MEIF-CT-2003-501940.

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Chiodo, A. Quantitative Néron theory for torsion bundles. manuscripta math. 129, 337–368 (2009). https://doi.org/10.1007/s00229-009-0261-8

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  • DOI: https://doi.org/10.1007/s00229-009-0261-8

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