N\'eron models, minimal models, and birational group actions

TL;DR

This paper investigates the relationship between Néron models and minimal models of Abelian varieties over Dedekind domains, establishing conditions for regular actions and the openness of Néron models within minimal models.

Contribution

It proves that the identity component of the Néron model acts regularly on minimal models and introduces a rigidity criterion for birational group actions.

Findings

The identity component of the Néron model acts regularly on minimal models.

Conditions under which the Néron model is an open subset of a minimal model.

A new rigidity criterion for birational group actions.

Abstract

Let $A_K$ be an Abelian variety over the quotient field of a Dedekind domain $R$. We show that the identity component of the N\'eron model of $A_K$ acts regularly on any minimal model of $A_K$ over $R$, and discuss when the N\'eron model is an open subset of a minimal model. The main technical result is the rigidity of small modifications, leading to a regularity criterion for birational group actions.