Semi-factorial nodal curves and N\'eron models of jacobians

TL;DR

This paper characterizes semi-factoriality of families of nodal curves over discrete valuation rings using dual graph combinatorics and extends Néron model constructions to singular generic fibres.

Contribution

It provides a necessary and sufficient combinatorial condition for semi-factoriality in nodal cases and extends Néron model construction to singular generic fibres.

Findings

A criterion for semi-factoriality based on dual graph combinatorics.

Performing a specific blow-up yields a semi-factorial model.

Extension of Néron model construction to singular generic fibres.

Abstract

A family of curves over a discrete valuation ring is called semi-factorial if every line bundle on the generic fibre extends to a line bundle on the total space. In the nodal case, we give a sufficient and necessary condition for semi-factoriality, in terms of combinatorics of the dual graph of the special fibre. In particular, we show that performing one blow-up centered at the non-regular closed points yields a semi-factorial model of the generic fibre. As an application, we extend Raynaud's construction of the N\'eron (lft)-model of the jacobian of the generic fibre of a family of nodal curves to the case where the generic fibre is singular.