Néron models of \(Pic^0\) via \(Pic^0\)

We provide a new description of the Néron model of the Jacobian of a smooth curve \(C_K\) with stable reduction \(C_R\) on a discrete valuation ring \(R\) with field of fractions \(K\). Instead of the regular semistable model, our approach uses the regular twisted model, a twisted curve in the sense...

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Bibliographic Details
Published in:arXiv.org 2015-09
Main Author: Chiodo, Alessandro
Format: Article
Language:English
Subjects:
Bundles
Combinatorial analysis
Fields (mathematics)
Subgroups
Online Access:Get full text
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Summary:We provide a new description of the Néron model of the Jacobian of a smooth curve \(C_K\) with stable reduction \(C_R\) on a discrete valuation ring \(R\) with field of fractions \(K\). Instead of the regular semistable model, our approach uses the regular twisted model, a twisted curve in the sense of Abramovich and Vistoli whose Picard functor contains a larger separated subgroup than the usual Picard functor of \(C_R\). In this way, after extracting a suitable \(l\)th root from the uniformizer of \(R\), the pullback of the Néron model of the Jacobian represents a Picard functor \(Pic^{0,l}\) of line bundles of degree zero on all irreducible components of a twisted curve. Over \(R\), the group scheme \(Pic^{0,l}\) descends to the Néron model yielding a new geometric interpretation of its points and new combinatorial interpretations of the connected components of its special fibre. Furthermore, by construction, \(Pic^{0,l}\) is represented by a universal group scheme \(Pic^{0,l}_{g}\) of line bundles of degree zero over a smooth compactification \(\overline{M}_g^l\) of \(M_g\) where all Néron models of smoothings of stable curves are cast together after base change.
ISSN:2331-8422