TITLE: Some arithmetic terminology: "universal domain", "specialization", "Chow point"
QUESTION [9 upvotes]: 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...)

REPLY [3 votes]: Igusa uses Weil's language, in a modified/enhanced version that deals
with reduction mod primes.  (My memory is that there is a paper of Shimura from the 50s that develops this language.)  It's not so easy to read it carefully, unfortunately.
Chow's method for constructing Jacobians (explained in his paper in the American Journal
from the 50s, again if memory serves) is, I think, as follows:
take $Sym^d C$ for $d > 2g - 2$. The fibres of the map $Sym^d C \to Pic^d(C)$
are then projective space of uniform dimension (by Riemann--Roch), and so it is not so hard to quotient out
by all of them to construct $Pic^d(C)$ (for $d > 2g - 2$), and hence to construct the Jacobian.  (I hope that I'm remembering correctly here; if not, hopefully someone will correct me.)
I think that this should be contrasted with the more traditional method of considering $Sym^g C$, which maps birationally to $Pic^g(C)$, i.e. with fibres that are generically points, but which has various exceptional fibres of varying dimensions, making it harder to form the quotient, thus inspiring in part Weil's "group chunk" method where he uses the group action
to form the quotient (in an indirect sort of way), and consequently loses some control of the
situation (e.g. he can't show that the Jacobian so constructed is projective).   I should
also say that it's been a long time since I looked at this old 1950s literature, and I'm not 
completely confident that I understand its thrust (i.e. I'm not sure what was considered easy and what was considered hard, and what was considered new and innovative in various papers as contrasted to what was considered routine), so take this as a very rough guide only.