Relative compactifications of semiabelian Néron models, I

Abstract

Let $R$ be a complete discrete valuation ring, $k(\hat{I}\cdot )$ its fraction field, $S:=\mathrm{Spec}R$, $(G_{\hat{I}}\cdot ,{\mathcal{L}}_{\hat{I}}\cdot )$ a polarized abelian variety over $k(\hat{I}\cdot )$ with ${\mathcal{L}}_{\hat{I}}\cdot$ ample cubical and $\mathcal{G}$ the Néron model of $G_{\hat{I}}\cdot$ 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 ($\hat{I}\pm$) $P$ is Cohen-Macaulay with codim${}_P(P\setminus \mathcal{G})=2$ and (${\hat{I}}^2$) $\mathcal{N}$ is ample invertible with ${\mathcal{N}}_{|\mathcal{G}}$ cubical and ${\mathcal{N}}_{\hat{I}}\cdot ={\mathcal{L}}_{\hat{I}}^{\otimes n}\cdot$ for some positive integer $n$.

77 pages, 2 figures