N\'eron models of Jacobians over bases of arbitrary dimension

TL;DR

This paper constructs Néron models of Jacobians over arbitrary-dimensional bases by analyzing nodal models of curves, providing explicit descriptions and criteria for separation.

Contribution

It introduces a method to build Néron models of Jacobians over complex bases using nodal models and describes their local and combinatorial properties.

Findings

Blowing up nodal models preserves their nodal structure.

Constructed Néron models as quotients of Picard spaces.

Provided a criterion for the Néron model to be separated.

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 N\'eron 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 N\'eron model to be separated.