Compactification of degenerate abelian schemes over a regular divisor

Abstract

We consider a semiabelian scheme $G$ over a regular base scheme $S$, which is generically abelian, such that the points of the base where the scheme is not abelian form a regular divisor $S_0$. We construct a compactification of $G$, that is a proper flat scheme $P$ over the base scheme, containing $G$ as a dense open set, such that $P_{S_0}$ is a divisor with normal crossings in $P$. We also show that given an isogeny between two such semiabelian schemes, we can construct the compactifications so that the isogeny extends to a morphism between the compactifications.