I'm looking for a family of abelian varieties $A\rightarrow S$ over a base that is finite type over $\mathbb{Q}$ (or $\mathbb{Z}$) that is "comprehensive" in the following sense: for every characteristic zero field $K$ and every abelian variety $B/K$ (of specified dimension $g$ and polarization degree $d^2$), there exists a $K$-point $s$ of $S$ such that the fiber $A_s$ is isomorphic to $B$. For instance, the family $E\subset \mathbb{P}^2\times \operatorname{Spec}(\mathbb{Z}[a,b][1/(4a^3+27b^2)])$ cut out by $y^2=x^3+ax+b$ does the trick for elliptic curves. I am aware that there is a "universal abelian scheme" $A_{g,d}$ quasi-projective over $\operatorname{Spec}(\mathbb{Z})$ which is the coarse moduli space for the DM stack $\mathcal{A}_{g,d}$, but (if I understand correctly) this only obtains geometrically irreducible fibers over $\operatorname{Spec}(\mathbb{Z}[1/d, \zeta_d])$ so is ruled out for our purposes since the $K$-points of the base only come from fields with a primitive $d$-th root of unity. (EDIT: I think I am not understanding correctly. Upon further review it seems like my remark about geometric irreducibility actually applies to the stack $\mathcal{A}_{g,d,n}$ rather than to $\mathcal{A}_{g,d}$ so perhaps this coarse moduli space $A_{g,d}$ could work after all.)
I think one could get ahold of such a family as follows: every such abelian variety can be embedded in projective space and these subvarieties should have the same Hilbert polynomial (calculated with respect to a certain line bundle). If $Z$ is the universal family corresponding to the relevant Hilbert scheme, then $\mathit{Hom}(Z\times Z, Z)$ can be realized as an open subscheme of a Hilbert scheme by a theorem of Grothendieck (in Bourbaki no. 221). Asking that the relevant diagrams commute is then a closed condition on this $\mathit{Hom}$ space. This is rough sketch and I'm not fluent enough in the details to see whether or not I'm cheating. Any advice would be greatly appreciated!
EDIT 2: Jason Starr gives an excellent (and affirmative) answer below, but his methods are well beyond my current understanding. In an effort to further my own understanding, I further developed my rough outline above into a full-fledged construction. Here are the details: Consider the functor $\operatorname{Hilb}^*(n):(Schemes)^{op}\rightarrow Sets$ given by $$T\mapsto \{(Y\subset \mathbb{P}^n_T, s: T\rightarrow Y) : Y\subset \mathbb{P}^n_T\text { is closed }, Y/T\text{ is flat, and } s\text{ is a section}\}.$$ This functor is represented by the universal object $Z/\operatorname{Hilb}(n)$, which is quasi-projective over $\operatorname{Spec}(\mathbb{Z})$ (this is easy to verify). For any fixed rational polynomial $\Phi$, the universal object $Z_{\Phi}/\operatorname{Hilb}(n,\Phi)$ represents the functor $\operatorname{Hilb}^*(n,\Phi)$ where one further specifies that $Y\subset \mathbb{P}^n_T$ has Hilbert polynomial $\Phi$. It is known that $Z_{\Phi}$ is projective over $\operatorname{Spec}(\mathbb{Z})$. Now, given $A/K$ of dimension $g$ and a polarization $\phi:A\rightarrow \check{A}$ of degree $d^2$, one can construct a certain very ample line bundle over $A$ which induces an embedding $A\hookrightarrow \mathbb{P}^n_K$ for which the Hilbert polynomial of $A$ is some $\Phi$ determined entirely by $g$ and $d$. We thus obtain a map $\operatorname{Spec}(K)\rightarrow\operatorname{Hilb}(n,\Phi)$, which factors through the universal object $Z_{\Phi}$ since abelian varieties come with sections. Let $(Z'/Z_{\Phi}, \varepsilon: Z_{\Phi}\rightarrow Z')$ be the universal object of $\operatorname{Hilb}^*(n,\Phi)$. After throwing away the singular locus, Theorem 6.14 from GIT ensures that $Z'/Z_{\Phi}$ is an abelian scheme with the specified section $\varepsilon$ serving as the identity. Abelian varieties are smooth, so I didn't accidentally throw away the fibers I care about when restricting to the smooth locus. The base of this family is quasi-projective (whence finite type) over $\operatorname{Spec}(\mathbb{Z})$.
Best Answer
Welcome new contributor. The idea of such "comprehensive" families goes back very far. These were studied by Amitsur under the name "generic splitting varieties", primarily in connection with problems in the theory of algebraic groups (with further connections to K-theory, Galois cohomology, etc.).
Definition (cf. Théorème 6.2 of [LM-B]) For a stack $\mathcal{X}$, a separated, smooth $1$-morphism from a scheme, $p:X\to \mathcal{X}$, is a generic splitting variety if for every field $K$ and for every $1$-morphism $\zeta:\text{Spec}(K) \to \mathcal{X}$, there exists a morphism of schemes $z:\text{Spec}(K)\to X$ such that $p\circ z$ is $2$-equivalent to $\zeta$.
By Théorème 6.2 of Laumon and Moret-Bailly [LM-B], an algebraic stack has a generic splitting variety if it satisfies a version of Grothendieck's "Main Theorem of Zariski".
[LM-B]
MR1771927
Laumon, Gérard; Moret-Bailly, Laurent
Champs algébriques.
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 39.
Springer-Verlag, Berlin, 2000.
The morphism constructed in [LM-B] is typically not quasi-compact.
The question studied by de Jong and myself is a bit sharper: for an algebraic stack $\mathcal{X}$ of finite type over a Noetherian scheme $T$, can we find generic splitting varieties where $X$ is a dense open in a projective $T$-scheme $\overline{X}$ whose closed complement $\overline{X}\setminus X$ has arbitrarily large codimension. This is useful for reducing the types of questions studied by Amitsur (and many others) to the case where the base scheme of the family is also projective over $T$, i.e., "avoiding the discriminant", "eliminating the branch locus", etc. It is also useful for defining "heights" for $K$-points of stacks. For a choice of ample invertible sheaf $\mathcal{O}(1)$ on $\overline{X}$, we can define the height of $\zeta$ to be the minimal $\mathcal{O}(1)$-height of a lift $z$ of $\zeta$.
For global quotient stacks $[V/G]$ with $V$ projective over $T$ and $G$ an affine group scheme over $T$, de Jong and constructed such generic splitting varieties using methods of Geometric Invariant Theory, cf. "Almost proper GIT-stacks and discriminant avoidance." This was slightly generalized in my work with Chenyang Xu, "Rational points of rationally simply connected varieties over global function fields."