[Paper Review] A criterion for existence of Néron models of jacobians

Giulio Orecchia|arXiv (Cornell University)|Jun 14, 2018

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition—toric additivity—for the existence of Néron models of jacobians of nodal curves over regular schemes of dimension greater than one. Toric additivity, defined via the equality of Betti numbers of dual graphs of fibers, ensures that the toric rank of the jacobian at a singular fiber equals the sum of toric ranks at generic points of boundary components. The key result is that a Néron model exists if and only if the family is toric-additive, resolving a long-standing question in higher-dimensional Néron model theory for jacobians.

ABSTRACT

Néron models of abelian varieties do not necessarily exist if the base $S$ has dimension higher than 1. We introduce a new condition, called toric additivity, on a family of smooth curves having nodal reduction over a normal crossing divisor $D\subset S$. The condition is necessary and sufficient for existence of a Néron model of the jacobian of the family; it depends only on the Betti numbers of the dual graphs of the fibres of the family, or on the toric ranks of the fibres of the jacobian.

Motivation & Objective

• To resolve the open problem of when Néron models exist for jacobians of nodal curves over regular schemes of dimension >1.
• To identify a precise, intrinsic condition on the family of curves that guarantees the existence of a Néron model for its jacobian.
• To clarify the relationship between toric additivity, alignment (Holmes' condition), and the regularity of the total space of the curve family.
• To establish that toric additivity is both necessary and sufficient for Néron model existence under mild hypotheses (e.g., excellent base scheme).

Proposed method

• Introduce the new condition of toric additivity, defined by equality of the first Betti number of the dual graph of a singular fiber and the sum of Betti numbers of the dual graphs of its components at the boundary.
• Use the generalized jacobian Pic^0_{C/S} as a functorial, semi-abelian model that captures the toric part of the jacobian, enabling the condition to be checked locally.
• Prove that toric additivity is stable under étale base change and blow-ups of the total space, preserving the jacobian's Pic^0 structure.
• Show that toric additivity implies the existence of a regular model after a suitable blow-up, allowing application of known results on Néron model existence for regular total spaces.
• Use descent techniques and the fact that toric additivity descends along étale covers to reduce the problem to the case where the total space is regular.
• Leverage Holmes’ alignment condition and its equivalence to toric additivity in the regular case to prove the converse direction.

Experimental results

Research questions

• RQ1 Under what conditions does the jacobian of a nodal curve over a regular scheme of dimension >1 admit a Néron model?
• RQ2 How does the toric rank of the jacobian at a singular fiber relate to the toric ranks at the generic fibers of the boundary components?
• RQ3 Is there a combinatorial condition on the dual graphs of fibers that characterizes Néron model existence for jacobians in higher dimensions?
• RQ4 Does toric additivity imply the existence of a regular model after base change or blow-up?
• RQ5 Is toric additivity equivalent to Holmes’ alignment condition when the total space is regular?

Key findings

• Toric additivity is a necessary and sufficient condition for the existence of a Néron model of the jacobian over a regular scheme S, provided S is excellent.
• The condition is equivalent to the equality h₁(Γ) = h₁(Γ₁) + … + h₁(Γₙ), where Γ is the dual graph of the singular fiber and Γᵢ are the dual graphs of the fibers over the generic points of the boundary components.
• Toric additivity is stable under étale base change and blow-ups of the total space, preserving the jacobian's Pic^0 structure.
• If the total space of the curve family is regular, toric additivity is equivalent to Holmes’ alignment condition.
• The jacobian admits a Néron model over S if and only if the family is toric-additive, even when the total space is not regular.
• The condition is open on S, so there exists a maximal open subset V ⊂ S over which the jacobian admits a Néron model.