Let $S$ be a locally noetherian scheme, $C$ the category of proper smooth $S$-schemes with geometrical connected fibres and $C\_\*$ the category of pointed objects of $S$, i.e. objects of $C$ together with a morphism $S \to C$. Also denote $A$ the category of abelian schemes over $S$. There is a well-known rigidity result stating that a pointed morphism between $X,Y \in A$ is already a group morphism. In other words, the inclusion functor


$A \to C\_\*$


is fully faithful. Is there a nice description for the image? In other words, which purely scheme-theoretic properties do abelian schemes have and are there enough to characterize them? For example, $X \in A$ is "homogeneous".



scheme-theoretic description of abelian schemes
How about smooth proper morphisms $X \to S$ with connected fibers, a section $S \to X$, such that the sheaf of Kähler differentials $\Omega\_{X/S}$ is a pullback from $S$, and such that the group scheme $\underline{\rm Aut}\_S X$ acts transitively on the fibers? The essential point is that the hypothesis on the differentials insures that no geometric fiber can contain a rational curve, so no affine algebraic group can act non-trivially. The result should follow from Chevalley's structure theorem for algebraic groups, with some fairly standard arguments (I haven't checked the details, though).


