Relative compactifications of semiabelian N\'eron models, I

TL;DR

This paper constructs a unique relative compactification of totally degenerate semiabelian Néron models over a discrete valuation ring, ensuring specific geometric and line bundle properties.

Contribution

It introduces a novel, unique relative compactification with Cohen-Macaulay structure and ample line bundle conditions for semiabelian Néron models.

Findings

Existence of a unique relative compactification with Cohen-Macaulay property.

The compactification has a boundary of codimension two.

The line bundle on the compactification is ample and cubical.

Abstract

Let $R$ be a complete discrete valuation ring, $k(\eta)$ its fraction field, $S:={\rm Spec} R$, $(G_{\eta},\mathcal{L}_{\eta})$ a polarized abelian variety over $k(\eta)$ with $\mathcal{L}_{\eta}$ ample cubical and $\mathcal{G}$ the N\'eron model of $G_{\eta}$ over $S$. Suppose that $\mathcal{G}$ is totally degenerate semiabelian over $S$. Then there exists a (unique) relative compactification $(P,\mathcal{N})$ of $\mathcal{G}$ such that ($\alpha$) $P$ is Cohen-Macaulay with codim$_P(P\setminus\mathcal{G}) = 2$ and ($\beta$) $\mathcal{N}$ is ample invertible with $\mathcal{N}_{|\mathcal{G}}$ cubical and $\mathcal{N}_{\eta}=\mathcal{L}^{\otimes n}_{\eta}$ for some positive integer $n$.