A Néron model of the universal jacobian
Annales Henri Lebesgue, Volume 4 (2021), pp. 1727-1766.

Metadata

KeywordsNéron models, jacobians, moduli of curves

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 ${\stackrel{˜}{ℳ}}_{g,\phantom{\rule{0.166667em}{0ex}}n}\to {\overline{ℳ}}_{g,\phantom{\rule{0.166667em}{0ex}}n}$ after which a Néron model ${N}_{g,\phantom{\rule{0.166667em}{0ex}}n}/{\stackrel{˜}{ℳ}}_{g,\phantom{\rule{0.166667em}{0ex}}n}$ 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 ${\stackrel{˜}{ℳ}}_{g,\phantom{\rule{0.166667em}{0ex}}n}\to {\overline{ℳ}}_{g,\phantom{\rule{0.166667em}{0ex}}n}$ is separated and relatively representable. The Néron model ${N}_{g,\phantom{\rule{0.166667em}{0ex}}n}/{\stackrel{˜}{ℳ}}_{g,\phantom{\rule{0.166667em}{0ex}}n}$ 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 ${\stackrel{˜}{ℳ}}_{g,\phantom{\rule{0.166667em}{0ex}}n}$.

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