A Néron model of the universal jacobian

Abstract

The jacobian of the universal curve over ${\mathcal{M}}_{g,n}$ is an abelian scheme over ${\mathcal{M}}_{g,n}$. Our main result is the construction of an algebraic space $\beta :{\tilde{\mathcal{M}}}_{g,n}\to {\bar{\mathcal{M}}}_{g,n}$ over which this jacobian admits a Néron model, and such that $\beta$ is universal with respect to this property. We prove certain basic properties, for example that $\beta$ is separated, locally of finite presentation, and satisfies a certain restricted form of the valuative criterion for properness. In general, $\beta$ is not quasi-compact. We relate our construction to Caporaso's balanced Picard stack ${\mathcal{P}}_{d,g}$.

36 pages. Major revision. Notation completely revised. An important error in the old lemma 6.2 is corrected. New material added making the construction more explicit in the universal case, discussing the pullback of the universal Neron-model admitting morphism, and showing that it is smooth over $\mathbb{Z}$