Models of Jacobians of curves

Abstract

We show that the Jacobians of prestable curves over toroidal varieties always admit Néron models. These models are rarely quasi-compact or separated, but we also give a complete classification of quasi-compact separated group-models of such Jacobians. In particular we show the existence of a maximal quasicompact separated group model, which we call the saturated model, which has the extension property for all torsion sections. The Néron model and the saturated model coincide over a Dedekind base, so the saturated model gives an alternative generalisation of the classical notion of Néron models to higher-dimensional bases; in the general case we give necessary and sufficient conditions for the Néron model and saturated model to coincide. The key result, from which most others descend, is that the logarithmic Jacobian of [MW18] is a log Neron model of the Jacobian.

Logarithmic intersections of double ramification cycles

“…Given such (C/S, P, α, L), the map S → J given by L(α) is an extension of σ, so by the universal property of Jac ♦ we obtain a map from the stack of such quadruples to Jac ♦ . To show this is an isomorphism, we may work locally (so assume C/S to be nuclear in the sense of [HMOP20], and smooth over a dense open of S), then P can be taken trivial, and it is enough to show that the extension of σ is given by a PL function; but this follows from [MW20, Corollary 3.6.3].…”

“…After applying the above lemma we will show that PL functions always extend. We start by considering the case where the base S is very small (nuclear in the sense of [HMOP20]), after which we will glue to a global solution. Proof.…”

The moduli stack of enriched structures and a logarithmic compactification

“…Remark 4.11. The notion of r-richness is connected to, but slightly different both from the notion of being r-aligned from [Hol15] and from the notion of being log aligned from [HMOP20]. The difference from being r-aligned, except for the important fact that being r-aligned is a condition of a scheme instead of a log scheme, is mainly that for r-aligned one requires the labels in a circuit-connected component to all be of the form λ i a for some 1 ≤ λ i ≤ r, instead of requiring the labels in a cut to be of the form µ i a for some 1 | µ i | r. The reason for cuts is that, while circuit-connected components can split when contracting edges, cuts do behave nicely under contraction (cf.…”

“…We will prove that ARLC r is a log blowup of LC using nuclear charts, as defined in [HMOP20]. We start with a few general lemmas on blowups, and then, given a nuclear chart U of LC, write down an explicit ideal of the characteristic monoid such that U × LC ARLC r is exactly this blowup of U in this ideal.…”

“…Definition 6.10. A nuclear chart is a nuclear curve of MLC (see Definition 3.39 of [HMOP20]). This automatically comes with a strict étale map to LC.…”