# Is a semiabelian algebraic space a scheme?

Let $S$ be a scheme and let $A$ be a commutative separated smooth $S$-group algebraic space of finite presentation each of whose geometric fibers is an extension of an abelian variety by a torus. Is $A$ necessarily a scheme?

If the geometric fibers are assumed to be abelian varieties (so that $A$ is necessarily proper), then the answer is positive, see Chai, Faltings "Degeneration of abelian varieties", Theorem 1.9 on page 5. The answer is positive also in the case when $S$ is a Dedekind scheme: this is proved in Anantharaman's thesis (respectively, already by M. Artin if $S$ is the spectrum of a field). Is the answer positive in general?

• Ask Kai-Wen Lan; problems like this may arise in the construction of compactifications of PEL moduli schemes. By the method in Faltings-Chai, the general case should reduce to $S$ normal, connected, and finite type over $\mathbf{Z}$. By normality, the semi-abelian algebraic space is functorial in its generic fiber (Faltings sketches this for schemes early in his Mordell paper, but the argument works for algebraic spaces), so it suffices to produce some semi-abelian $S$-scheme with the same generic fiber. This one may hope to with moduli schemes, at least etale-locally on $S$. – user27920 Aug 10 '14 at 20:33
• Maybe the answers for mathoverflow.net/questions/8918 can help? – Matthias Wendt Aug 12 '14 at 16:57