[Paper Review] Compactified Jacobians as Mumford models

Karl Christ, Sam Payne+1 more|arXiv (Cornell University)|Dec 8, 2019

Algebraic Geometry and Number Theory21 references4 citations

TL;DR

This paper establishes that relative compactified Jacobians of one-parameter smoothings of nodal curves are Mumford models of the generic fiber's Picard variety, constructed via polytopal decompositions of the skeleton. The key result is that each compactified Jacobian corresponds to a specific decomposition tied to stability data, with the degree-g case uniquely realized by the tropical break divisor decomposition.

ABSTRACT

We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g there is a unique compactified Jacobian encoding slop stability, and it is induced by the tropical break divisor decomposition.

Motivation & Objective

• To establish a precise correspondence between relative compactified Jacobians and Mumford models in the context of one-parameter smoothings of nodal curves.
• To describe the polytopal decompositions of the skeleton that realize these compactified Jacobians, using auxiliary stability data.
• To show that the degree-g compactified Jacobian encoding slope stability corresponds uniquely to the tropical break divisor decomposition.
• To unify classical constructions of compactified Jacobians with non-archimedean analytic uniformization via Mumford models.
• To prove that the special fiber of the compactified Jacobian is isomorphic to the quotient of a formal model by a lattice, confirming the Mumford model structure.

Proposed method

• The authors use non-archimedean analytic uniformization to construct Mumford models from polytopal decompositions of the skeleton of the analytification of the Picard variety.
• They associate each compactified Jacobian to a decomposition of the skeleton that parametrizes combinatorial types of φ-polystable sheaves, with multidegrees and node sets as invariants.
• The decomposition is shown to generalize the Namikawa decompositions for metric graphs, with cells corresponding to tropical divisors on the dual graph with additional points at stable nodes.
• The proof relies on constructing two formal models: one from the Mumford uniformization and one from the compactified Jacobian, then proving their canonical isomorphism via affinoid decomposition and tropicalization.
• The key technical step involves showing that the generic fibers of the formal models agree on affine opens by matching isomorphism classes of line bundles extending to torsion-free sheaves with tropical equivalence classes.
• The isomorphism is glued across the atlas and descends to an algebraic isomorphism, leveraging unique algebraizability of formal completions for projective schemes.

Experimental results

Research questions

• RQ1 How can relative compactified Jacobians of one-parameter smoothings of nodal curves be realized as Mumford models?
• RQ2 What polytopal decomposition of the skeleton corresponds to a given numerical polarization φ in the compactified Jacobian construction?
• RQ3 Is there a unique compactified Jacobian in degree g that realizes slope stability, and how is it related to tropical geometry?
• RQ4 How do the moduli of φ-polystable sheaves on the special fiber relate to the combinatorial types in the skeleton decomposition?
• RQ5 Can the formal model of the compactified Jacobian be canonically identified with the Mumford model via uniformization?

Key findings

• The compactified Jacobian $\overline{J}_{\mathcal{X}}(\phi)$ is isomorphic to a Mumford model of $\mathrm{Pic}^d(\mathcal{X}_K)$, constructed from a polytopal decomposition of the skeleton of the analytification.
• The decomposition corresponding to $\overline{J}_{\mathcal{X}}(\phi)$ parametrizes tropical divisors with multidegree $\underline{d}$ on vertices and one additional point on edges corresponding to nodes in the support $S$ of the sheaf.
• In degree $g$, the unique compactified Jacobian realizing slope stability is induced by the tropical break divisor decomposition of the skeleton.
• The inclusion of the generic fiber $\mathrm{Pic}^d(\mathcal{X}_K)$ into $\overline{J}_{\mathcal{X}}(\phi)$ is toroidal, meaning it is étale locally isomorphic to a split torus in a toric variety.
• The special fiber $\overline{J}_X(\phi)$ is isomorphic to the quotient of a formal model $\mathfrak{E}$ by a lattice $\Lambda$, confirming the Mumford model structure.
• The formal models arising from the Mumford uniformization and from the compactified Jacobian are canonically isomorphic, leading to an algebraic isomorphism between the Mumford model and the compactified Jacobian.