Separated quotients of Picard schemes

TL;DR

This paper establishes conditions for the existence of Néron models of Jacobians in semistable families, introducing alignment as a key concept for separated models with Néron properties.

Contribution

It introduces the notion of alignment for semistable morphisms and characterizes when Jacobians admit separated Néron models over arbitrary bases.

Findings

Aligned semistable morphisms have Jacobians with separated Néron models.

The paper provides necessary and sufficient conditions for Néron model existence.

Smoothness of the Picard scheme along the unit section implies alignment.

Abstract

We give some necessary and sufficient conditions for the existence of N\'{e}ron models of jacobians of semistable morphisms of arbitrary relative dimension over base schemes of arbitrary dimension. To do this, we introduce a notion of alignment for semistable morphisms over any regular base scheme, and show that the jacobian of an aligned projective semistable morphism admits a separated model with the N\'{e}ron mapping property. When the Picard scheme is smooth over the base scheme along its unit section we show that the converse holds.