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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".

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The purely scheme-theoretic properties that Abelian schemes have which characterise the image are: (insert definition of an abelian scheme here, i.e. smooth proper, group structure, geom conn fibres). What more are you asking for? –  Kevin Buzzard Apr 28 '10 at 18:12
I'm asking for a char. which does not involve a group multiplication. For example, how can I decide whether $\mathbb{P}^1_S$ is an abelian scheme? This is just an example. –  Martin Brandenburg Apr 28 '10 at 18:35

1 Answer 1

up vote 6 down vote accepted

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).

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Triviality of $\Omega_{X/S}$ disposes of $\mathbb P^1$, at least! –  Mariano Suárez-Alvarez Apr 28 '10 at 19:50
Nice! By Theorem 6.14 in GIT (for which the projective hypothesis relaxes to properness by using Artin's results on Hilbert and Hom functors via algebraic spaces instead of schemes), the abelian scheme structure exists (uniquely) if it does so on geometric fibers. Thus, the proposed criterion above reduces to case when $S$ is spectrum of alg. closed field, in which case it follows by the suggested argument (using rational curves and Chevalley's theorem) provided we modify the hypothesis on Aut's to involve the identity component of the Aut-scheme on fibers (still geometric criterion). –  BCnrd Apr 28 '10 at 20:15
By a transitive action I mean that the morphism $\underline{\rm Aut}_S X \to X$ coming from the section is scheme-theoretically surjective. Since the fibers are connected, this implies that the connected component of the identity must dominate each fiber, and this is enough to conclude. –  Angelo Apr 28 '10 at 21:05
If the generic points of S have residue characteristic 0 then perhaps the condition on Aut_S X is unnecessary since any connected smooth proper variety over a field of characteristic 0 with trivial tangent bundle (and a rational point) is an abelian variety. This is not true in positive characteristics, but does suggest that some weaker condition might suffice. –  ulrich Apr 29 '10 at 11:09
@unknown: Could you please give an example of failure in positive characteristic, i.e. an example of a connected, smooth, proper scheme $X/k$ with trivial tangent bundle and a $k$-point, where $k$ is a field of positive characteristic, such that $X/k$ doesn't have a group law? –  Thanos D. Papaïoannou Apr 29 '10 at 22:23

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