Universal Néron models for Jacobians of curves with marked points (Q1680749)

This paper studies the problem of constructing Néron models for families of curves with marked points. The question can be formulated more precisely as follows. Let \(\mathfrak{M}_{g,n}\) be the moduli stack of reduced curves of genus \(g\) with \(n\) distinct marked points. Is there an algebraic stack \(\mathfrak{N}_{g,n}\) endowed with a map onto \(\mathfrak{M}_{g,n}\) such that \(\mathfrak{N}_{g,n}\) is a universal Néron model for the Jacobians of families of curves in \(\mathfrak{M}_{g,n}\)? Moreover, can one describe a modular compactification of \(\mathfrak{N}_{g,n}\) over \(\mathfrak{M}_{g,n}\)? The author shows that the universal compactified Jacobians constructed in her previous work [``Compactifications of the universal Jacobian over curves with marked points'', Preprint, \url{arXiv:1509.06177}] gives a positive answer to the above question in the case of families of reduced curves with at least one section and such that the total space of the family is regular.