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Néron models of Jacobians over bases of arbitrary dimension

Thibault Poiret

TL;DR

This work extends the construction of Néron models of Jacobians to bases of arbitrary dimension by exploiting nodal models of a smooth curve and controlled blow-ups along sections. A local description of the Jacobian NM is provided as a quotient of the Picard space $ ext{Pic}^{ ext{tot }0}_{X/S}$ by the étale closure of the unit section, with refinements of nodal models guiding the global structure. The paper proves the existence of a Néron model for the Jacobian under a smooth-factorial, excellent base and develops a combinatorial criterion (strict alignment) that determines separatedness. The results connect to and extend previous criteria (alignment, toric-additivity) and furnish a modular, graph-theoretic viewpoint via refinements of dual graphs and blow-ups in sections.

Abstract

We work with a smooth relative curve $X_U/U$ with nodal reduction over an excellent and locally factorial scheme $S$. We show that blowing up a nodal model of $X_U$ in the ideal sheaf of a section yields a new nodal model, and describe how these models relate to each other. We construct a Néron model for the Jacobian of $X_U$, and describe it locally on $S$ as a quotient of the Picard space of a well-chosen nodal model. We provide a combinatorial criterion for the Néron model to be separated.

Néron models of Jacobians over bases of arbitrary dimension

TL;DR

This work extends the construction of Néron models of Jacobians to bases of arbitrary dimension by exploiting nodal models of a smooth curve and controlled blow-ups along sections. A local description of the Jacobian NM is provided as a quotient of the Picard space by the étale closure of the unit section, with refinements of nodal models guiding the global structure. The paper proves the existence of a Néron model for the Jacobian under a smooth-factorial, excellent base and develops a combinatorial criterion (strict alignment) that determines separatedness. The results connect to and extend previous criteria (alignment, toric-additivity) and furnish a modular, graph-theoretic viewpoint via refinements of dual graphs and blow-ups in sections.

Abstract

We work with a smooth relative curve with nodal reduction over an excellent and locally factorial scheme . We show that blowing up a nodal model of in the ideal sheaf of a section yields a new nodal model, and describe how these models relate to each other. We construct a Néron model for the Jacobian of , and describe it locally on as a quotient of the Picard space of a well-chosen nodal model. We provide a combinatorial criterion for the Néron model to be separated.

Paper Structure

This paper contains 27 sections, 40 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Néron models
  3. Models of Jacobians
  4. Notation
  5. Structure of the paper, main results
  6. Generalities about Néron models
  7. Definitions
  8. Base change and descent properties
  9. Group models with injective restriction map
  10. Nodal curves and dual graphs
  11. First definitions
  12. The local structure
  13. The dual graph at a geometric point
  14. Quasisplitness, dual graphs at non-geometric points
  15. Primality, local factoriality and base change
  16. ...and 12 more sections

Key Result

Corollary 1.1

Let $S$ be a scheme, $U\subset S$ a scheme-theoretically dense open subscheme, $N_U \to U$ a smooth, separated $U$-group algebraic space and $f\colon N \to S$ a smooth $S$-group model of $N_U$. Denote by $E$ the scheme-theoretic closure of the unit section in $N$ and by $E^{\mathrm{\acute{e}tale}}$ is exact. In particular, the quotient space $N/E^{\mathrm{\acute{e}tale}}$ is a smooth $S$-group mo

Theorems & Definitions (118)

  • Corollary 1.1: cf. Corollary \ref{['corollary:quotient_of_group_space_by_E^et_has_uniqueness_in_NMP']}
  • Theorem 1.2: cf. Theorem \ref{['theorem:NMs_Jac']}
  • Theorem 1.3: cf. Theorem \ref{['theorem:separatedness_of_NM_jac_iff_strictly_aligned']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6: The formation of Néron models is compatible with smooth base change
  • proof
  • ...and 108 more