[Paper Review] Models of Jacobians of curves

David Holmes, Samouil Molcho+2 more|arXiv (Cornell University)|Jul 21, 2020

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes the existence of Néron models for Jacobians of prestable curves over toroidal varieties, generalizing classical Néron models to higher-dimensional bases. It introduces the saturated model—a maximal quasi-compact, separated group model with the extension property for torsion sections—and proves that the logarithmic Jacobian is a log Néron model, unifying the theory via logarithmic geometry.

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 quasi-compact 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 \cite{Molcho2018The-logarithmic} is a log Neron model of the Jacobian.

Motivation & Objective

• To extend the classical notion of Néron models to higher-dimensional bases, where separated, quasi-compact models rarely exist.
• To classify all separated, quasi-compact group models of the Jacobian in terms of subgroups of the tropical Jacobian.
• To introduce and characterize the saturated model as the maximal quasi-compact, separated group model with the extension property for torsion sections.
• To establish a log geometric interpretation of the Néron model via the logarithmic Jacobian.
• To provide necessary and sufficient conditions for the coincidence of the Néron model and the saturated model in higher dimensions.

Proposed method

• Construct the Néron model as the strict logarithmic Picard functor sLPic⁰, which represents degree-zero logarithmic line bundles on a prestable curve over a log regular base.
• Define the tropical Jacobian as the quotient of the Néron model by its identity component, forming an étale group algebraic space over the base.
• Use the tropical Jacobian to classify quasi-finite open subgroups, which correspond bijectively to separated, quasi-compact group models of the Jacobian.
• Prove that the logarithmic Jacobian is a log Néron model by verifying the universal extension property for log morphisms.
• Establish the existence of the saturated model as the maximal such model via a saturation process in the Picard functor.
• Apply descent and representability techniques on the big site of schemes to prove representability of the relevant functors.

Experimental results

Research questions

• RQ1 Does a Néron model exist for the Jacobian of a prestable curve over a higher-dimensional toroidal base?
• RQ2 What are the necessary and sufficient conditions for the Néron model and the saturated model to coincide in higher dimensions?
• RQ3 How can separated, quasi-compact group models of the Jacobian be classified in terms of tropical invariants?
• RQ4 What is the role of logarithmic geometry in constructing and characterizing Néron models for higher-dimensional bases?
• RQ5 Can the saturated model be characterized as a universal extension for torsion sections?

Key findings

• The Néron model of the Jacobian of a prestable curve over a toroidal variety always exists and is quasi-separated.
• The logarithmic Jacobian sLPic⁰ is a log Néron model, providing a modular interpretation via logarithmic line bundles.
• The saturated model is the maximal quasi-compact, separated group model of the Jacobian and has the extension property for all torsion sections.
• There is a canonical bijection between quasi-finite open subgroups of the tropical Jacobian and separated, quasi-compact group models of the Jacobian.
• The Néron model and the saturated model coincide if and only if the curve satisfies a condition related to alignment of smoothing parameters.
• The strict tropical Jacobian sTPic⁰ is isomorphic to the component group of the Néron model over local Dedekind bases.