Abstract
Jacobians of degenerating families of curves are well-understood over 1-dimensional bases due to work of Néron and Raynaud; the fundamental tool is the Néron model and its description via the Picard functor. Over higher-dimensional bases Néron models typically do not exist, but in this paper we construct a universal base change ℳ ˜ g,n →ℳ ¯ g,n after which a Néron model N g,n /ℳ ˜ g,n of the universal jacobian does exist. This yields a new partial compactification of the moduli space of curves, and of the universal jacobian over it. The map ℳ ˜ g,n →ℳ ¯ g,n is separated and relatively representable. The Néron model N g,n /ℳ ˜ g,n is separated and has a group law extending that on the jacobian. We show that Caporaso’s balanced Picard stack acquires a torsor structure after pullback to a certain open substack of ℳ ˜ g,n .
Highlights
One would like a model of Jg, n over Mg, n which is proper and admits a group structure; properness guarantees that sections extend and intersection theory makes sense, and the group law gives additional structure to the resulting cycles and linear series
(1) Mg, n → Mg, n is an isomorphism over Mg, n; (2) the universal jacobian Jg, n admits a Néron model over Mg, n
The map Mg, n → Mg, n is not proper in general; this is forced upon us by condition (2) and the results of [Hol19], unless a Néron model already exists over Mg, n
Summary
Abstract. — Jacobians of degenerating families of curves are well-understood over 1dimensional bases due to work of Néron and Raynaud; the fundamental tool is the Néron model and its description via the Picard functor. Over higher-dimensional bases Néron models typically do not exist, but in this paper we construct a universal base change Mg, n → Mg, n after which a Néron model Ng, n/Mg, n of the universal jacobian does exist. This yields a new partial compactification of the moduli space of curves, and of the universal jacobian over it. — Les jacobiennes de dégénérescences de courbes au-dessus de bases de dimension 1 sont bien comprises grâce aux travaux de Néron et Raynaud ; l’outil fondamental est le modèle de Néron et sa description à l’aide du foncteur de Picard.
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