Relative compactifications of semiabelian N\'eron models, I

Let $R$ be a complete discrete valuation ring, $k(\eta )$ its fraction field, $S:=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R$, $(G_{\eta },{\mathcal{L}}_{\eta })$ 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 ${\mathcal{N}}_{\mathrm{\mid }\mathcal{G}}$ cubical and ${\mathcal{N}}_{\eta }={\mathcal{L}}_{\eta }^{\otimes n}$ for some positive integer $n$.