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In Lang's Survey on Diophantine Geometry, page 40, he said the following:

Let $F=k(Y)$ be a function field of variety $Y$ over the constant field $k$ and $X_F$ a non-singular projective variety over $F$. Then we may view $X_F$ as the generic fiber of a family, namely there exists a morphism $\pi : X \rightarrow Y$ such that the generic fiber is $X_F$. Then there exists a non-empty Zariski open $Y_0$ of $Y$ such that $\pi$ is smooth over $Y_0$. For each $y \in Y_{0}(K^a)$ we get a fiber $X_y$.

A rational point $P \in X_F(F)$ corresponds to a rational section $s_P: Y \rightarrow X$ and for $y \in Y_0$ the imbedding $\{ y\} \subset Y$ induces a point $s_P(y) \in X_y(k(y))$. The map $X_F(F) \rightarrow X_y(k(y))$ where $P \mapsto s_P(y)$ is called the specialization map. If $X=A$ is an abelian variety, then the specialization map is a homomorphism.

My questions are:

  1. For all $P\in X_F(F)$, all $s_P$ are just rational maps, why are they all defined at $y \in Y_{0}(K^a)$?

  2. Why is the specialization map a homomorphism when $X=A$ is an abelian variety?

Thank you.

Edited: So we have the answer (please refer to responses below) for 1 and 2 if $X$ is an abelian scheme over a smooth scheme $Y_0$. So is it true that given any abelian variety $X_F$, there exists an abelian scheme $\pi : X \rightarrow Y_0$ over $Y_0$ for some non-empty Zariski open smooth scheme $Y_0 \subset Y$, such that the generic fiber is $X_F$? The relevant context for this question is from Serre's Lectures on the Mordell-Weil Theorem, chapter 11, page 152 where he talks about Neron specialization theorem:

Let $k$ be a number field and $A$ an abelian variety over $K=k(T_1, \ldots, T_n)$. Since $K$ is the function field of $\mathbb{P}^n$ and $A$ is defined over $K$, $A$ comes from some abelian scheme $A_U$ over a non-empty open subset $U$ of $\mathbb{P}^n$.

So my question is: why does such abelian scheme $A_U$ exist? Thank you.

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    $\begingroup$ It is not true that $s_P$ is a genuine section in general, so the answer to question 1 is "they're not". But if $X$ is an abelian variety then Weil's Extension Theorem ensures that any rational map to $X$ from a $k$-variety is a morphism. But question 2 makes no sense at all since $X_F$ isn't an $F$-group in general. Perhaps if you provide some context/motivation for your question then it will become apparent how to give useful answers to these questions that are somehow each not well-posed. $\endgroup$ – user27920 Jun 12 '14 at 17:21
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    $\begingroup$ Are you perhaps just starting with an abelian variety over $F$ and then "spreading out" to an abelian scheme over a smooth dense open subset of $Y$? The algebraic geometry technique in Lang's book is poorly-suited for relative questions in algebraic geometry (though Lang et al. certainly got some mileage out of it before the whole framework was abandoned in the late 1950's). $\endgroup$ – user27920 Jun 12 '14 at 17:23
  • $\begingroup$ The case I'm interested in is $X_F$ is an algebraic variety over $F$. Let $Y_0$ be the Zariski open subset of $Y$ such that each fiber $X_y$ is an abelian variety for $y \in Y_0$. $\endgroup$ – wongpin101 Jun 12 '14 at 17:45
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    $\begingroup$ @wongpin101: I think you are missing a point in the definition of the specialization map. Let $X/S$ be an abelian scheme, where $S$ is integral, with field of fractions $F$. First assume that $S$ is the spectrum of a DVR. Then the valuative criterion of properness furnishes an isomorphism of groups $X(S) \to X_F(F)$. On the other hand, if $s$ is the special point of $S$, one has a morphism of groups (functoriality) $X(S)\to X_s(\kappa(s))$. These two properties give the specialization morphism $X_F(F)\to X_s(\kappa(s))$. $\endgroup$ – ACL Jun 13 '14 at 5:31
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    $\begingroup$ @wongpin101 (followed). This continues to work if $S$ is the spectrum of a Dedekind ring. In the general case, the morphism of groups $X(S)\to X_F(F)$ is injective but has no reason to be an isomorphism. This is already the case if $S$ is not normal. And if $S$ is normal, a morphism $Spec(F)\to X$ only extends outside of a codimension 2 subset in general. $\endgroup$ – ACL Jun 13 '14 at 5:33
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This is addressing the final question at the end of the edited version of the posted question, which finally gets to the heart of the matter (and without which the earlier parts of the question don't make sense, but after which the earlier parts of the question are all straightforward, so it is this final part which contains the real content).

The function field is the direct limit of the coordinate rings of non-empty affine opens. So one "just" has to apply a standard formalism for "spreading out" (or "denominator-chasing") from a finitely presented algebro-geometric situation over a limit ring to one over some stage of the limit process. This is exhaustively documented in EGA IV$_3$, sections 8, 9, 11 and IV$_4$, and is intuitively plausible in the affine setting but requires good technique beyond the affine case and especially for properties of geometric fibers (e.g., geometrically connected) and of morphisms (e.g., flatness, surjectivity). The main properties which are non-trivial to spread out in the circumstance of interest are geometric connectedness of fibers and flatness (or smoothness, depending on how you want to approach the latter in a relative setting).

It is best to figure out this particular elementary case on your own (using an ambient projective space to keep track of properness, and the Jacobian criterion to keep track of smoothness, using generic flatness appropriately).

But here is another way, via the general spreading-out principles. By regarding these open affines as an inverse system, apply EGA IV$_3$ 8.8.2(ii) to spread the abelian variety over the function field to a finitely presented scheme over a dense open inside the base. Then shrinking the open some more allows one to "spread out" the identity section and multiplication and inverse morphisms satisfying commutative group scheme axioms via 8.8.2(i), and to attain properness for the structural morphism by 8.10.5(xii), geometrically connected fibers by 9.7.7(i), and smoothness by IV$_4$, 17.7.8(ii). QED

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