Specialization Map of family of abelian varieties

Question

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.

Answer 1

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