As a non-connoisseur of arithmetic and arithmetic geometry, I would like to ask about some terminology, which meaning I haven't been able to find out on some books, nor on wikipedia, nor by google.
First,
What's a "universal domain" (of a given characteristic)?
What I knew is that, on a scheme, a (not necessarily closed) point x is called a specialization of a point y (which in turn is called a generization of x) if x lies on the topological closure of the sigleton {y}.
Does it make sense, in some context, to say that a given scheme (or subscheme of a fixed scheme) is the "specialization" of another one?
Suppose you are in the following context (that I will naively try to set). You are given a scheme $M$ over the integers, such that over points of $Spec \mathbb{Z}$ it has fibers that are algebraic varieties over the residue fields $\mathbb{F}_p$, $p\geq 0$, (or maybe over a "universal domain" of suitable characteristic?). Suppose also that it is kind of an "arithmetic moduli space" for e.g. curves of some genus so that closed points of its fiber "over p" parametrize curves of that genus in characteristic p.
In the above context, or in a similar one, does it make sense to say that "a curve $C'$ is a specialization of another curve $C$"? What about the assertion "the jacobian $J'$ is a specialization of the jacobian $J$"?
Also,
What's a "specialization over another specializaion"?
What's the "Chow method" to construct the jacobian of a nonsingular curve (of any characteristic)?
What's the "Chow point"? (I suppose it's a concept related to field extensions…)
Best Answer
Dear unkown,
I assume you must be trying to read something written long ago, and that you know about schemes. My earlier comment was perhaps overly optimistic: it's not clear that there's anyone here who knows Weil's Foundations of Algebraic Geometry. However, I found this article by Raynaud http://www.ams.org/notices/199908/fea-raynaud.pdf which should help.
Addendum Universal domains, which are big algebraically closed fields, were the crutch on which Weil's theory rested. However, as Pete Clark suggested, I think they still have some utility in post Weil algebraic geometry. I don't know about saturated models, but here is a more pedestrian explanation of why I think so. Given an algebraically closed field $k$, the composita of all function fields over it would lead to a universal domain $K\supset k$. So $K$ gives a convenient way to encode generic behaviour of all $k$-varieties.
To give an example where this sort of thing was useful, one can take a look at a paper of Bloch and Srinivas, Amer. J. Math 1983, where the hypothesis involved the Chow group of $0$-cycles on $X_K=X\otimes K$, where $X$ was a variety over $k$. This was really a way of packaging information about relative $0$-cycles on $(X\times Y)/Y$, for variable $Y$. In particular, their main result was obtained by applying this to the diagonal when $Y=X$.