A Néron model of the universal jacobian

Holmes, David

Annales Henri Lebesgue, Tome 4 (2021), pp. 1727-1766.

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. En général, sur des bases de dimension supérieure les modèles de Néron n’existent pas, mais dans cet article nous construisons un changement de base ${\widetilde{ℳ}}_{g,n}\to {\overline{ℳ}}_{g,n}$ qui est universel pour la propriété qu’un modèle de Néron $N_{g,n}/{\widetilde{ℳ}}_{g,n}$ de la jacobienne universelle existe. Ceci fournit une nouvelle compactification partielle de l’espace de modules des courbes et de la jacobienne universelle qui vit dessus. Le morphisme ${\widetilde{ℳ}}_{g,n}\to {\overline{ℳ}}_{g,n}$ est séparé et relativement représentable. Le modèle de Néron $N_{g,n}/{\widetilde{ℳ}}_{g,n}$ est séparé et possède une loi de groupe qui étend celle de la jacobienne. Nous montrons que le «  champ de Picard équilibré » de Caporaso acquiert une structure de torseur après changement de base à un certain sous-champ ouvert de ${\widetilde{ℳ}}_{g,n}$.

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 ${\widetilde{ℳ}}_{g,n}\to {\overline{ℳ}}_{g,n}$ after which a Néron model $N_{g,n}/{\widetilde{ℳ}}_{g,n}$ 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 ${\widetilde{ℳ}}_{g,n}\to {\overline{ℳ}}_{g,n}$ is separated and relatively representable. The Néron model $N_{g,n}/{\widetilde{ℳ}}_{g,n}$ 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 ${\widetilde{ℳ}}_{g,n}$.

Classification : 14K30, 14H10, 14H40
Mots clés : Néron models, jacobians, moduli of curves

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