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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.


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 $J$"?


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...)

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Some of the terminology you're asking about was more common in 1950's, with Weil's style of algebraic geometry, than it is now. I assume there is someone here more fluent in this language than me, so I won't attempt to answer most of your questions. However, I believe a universal domain is a field of infinite transcendence degree over the prime field. I assume that a Chow point is a point of the Chow variety parameterizes cycles of a fixed degree. –  Donu Arapura Aug 28 '10 at 16:32
I forgot to say that a universal domain is an algebraically closed field of .... –  Donu Arapura Aug 28 '10 at 16:41
What archaic reference are you trying to read, and for which results? –  BCnrd Aug 28 '10 at 20:09
OK, I'll bite. An example of such an archaic reference that I've wondered about are the 3 papers "Fiber systems of Jacobian varieties I,II,III" published in the Amer. Journal by Igusa between '56 and '59. They are written in the language that "Unknown" indicates. And since I'm no sort of expert in these matters, I don't know where to find the results in modern language. (I was led to them a while ago by an exercise in Serre's "Elliptic curves and ell-adic cohom" book: show for a suitable ell. curve over functn field k(t) with j-inv't t that the Gal. rep'n on Tate-group has image SL_2(Z_ell).) –  George McNinch Aug 29 '10 at 19:34
Dear unknown (google) [why isn't it ever "unknown (twitter)" or "unknown (yahoo)"...?]: OMG, that paper looks painful, working over $\mathbf{Z}$ from perspective of univ. domains. He "defines" $M_2$ and "constructs" it as affine scheme. Mumford defines/constructs $M_g$ over $\mathbf{Z}$ for all $g$ in GIT. Try Mumford's book "Curves & their Jacobians" & paper "Towards an enumerative geometry of moduli space of curves" (in Part III he gives conceptual pf that $M_2$ is affine, over alg. closed fields of char. 0), & find ideas over $\mathbf{Z}$ yourself or search refs to these on MathSciNet? –  BCnrd Sep 1 '10 at 3:22

3 Answers 3

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 $Y=X$.

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Thanks ! –  Qfwfq Aug 28 '10 at 17:39
Yes, I guessed that must have been an old fashioned terminology: I was looking for a translation into scheme theoretic language. –  Qfwfq Aug 28 '10 at 17:42
Dear Donu: The idea of replacing a $k$-variety $X$ with the $K$-variety $X_K$ for $K = k(X)$ comes up in theory of elliptic curves, and it is called "Hasse's trick" in Katz-Mazur (they used $X \times X$ viewed as an $X$-scheme via 2nd projection), and shows up in Silverman's book when he analyzes invariant differentials. Of course, it is implicit with many applications of Yoneda's Lemma via base change, or when making base change to acquire a section (the diagonal), etc. This all seems less drastic than "universal domains", but anyway perhaps the idea is due to Hasse rather than Weil? –  BCnrd Aug 29 '10 at 13:16
Dear Brian, it's a bit ironic that I find myself arguing in support of universal domains, since I could just as easily argue against their use. I'll concede that the Hasse trick (I didn't know it went back that far) doesn't require the full force of universal domains. To be honest, I don't know of any more compelling examples, which is not to say that don't exist. –  Donu Arapura Aug 29 '10 at 14:57
Dear Brian, could you please elaborate a bit on "it is implicit with many applications of Yoneda's Lemma via base change, or when making base change to acquire a section (the diagonal)"? –  Qfwfq Aug 30 '10 at 11:09

This is more of a comment than an answer, but I want to say that in my opinion this older terminology has not been completely superseded by scheme-theoretic language (only 98 percent superseded, or something like that).

In particular, the notion of a universal domain does not appear in scheme-theoretic algebraic geometry. The foundational work that this concept does in Weil's theory is rendered unnecessary by the theory of schemes, but I think the concept itself is still an important one. For instance, a universal domain is precisely a saturated model of the theory of algebraically closed fields, and this suggests both its geometric usefulness ("realization of types") and that we should look for other saturated models of various classes of fields.

Addendum: To clarify my point in the first paragraph: I am not claiming that scheme theory lacks the power to express any particular concept or definition employed by older schools of algebraic geometry. (The translation of older concepts into scheme-theoretic language is not always so straightforward. See for instance Brian Conrad's scheme-theoretic exposition of Chow's $K/k$-trace. But I think this is an instance of the nontriviality of simplification: there is no doubt that Brian's version is easier to read and understand than Chow's original.) Rather, what I am saying is that there are certain concepts that appear front-and-center in Weil-style algebraic geometry for foundational reasons but have an also non-foundational importance and usefulness. Most students of scheme-theoretic algebraic geometry are not taught to think in terms of universal domains and generic points (in the sense of Weil), whereas I believe this is a useful intuition. I have no doubt that this intuition can be expressed and even refined scheme-theoretically, but this does not seem to be as standard: most working algebraic geometers have not read straight through EGA!

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Pete, doesn't the theory of limits of schemes from EGA IV$_3$, sections 8ff. absorb everything ever done with universal domains (in the sense of exactly justifying all arguments where we massively increase or decrease the ground field to achieve certain aims, such as Lefschetz Principle or whatever)? Please give an example of something done with a "saturated model of the theory of alg. closed fields" not readily handled by the limit formalism of schemes. (In Appendix A in a recent paper of Breuillard-Green-Tao using ultrafilters, I think all of it can done via the limit formalism.) –  BCnrd Aug 29 '10 at 2:33
@B: I am not familiar with this part of EGA. I'm honestly not sure whether this is an embarrassing lacuna on my part or a justification of my point. But in any case, I think the idea of a saturated model -- a notion which is as natural and familar to a model theorist as Riemann-Roch is to an algebraic geometer -- provides an interesting point of contact between arithmetic algebraic geometry and model theory. I wonder whether the portion of EGA you mention develops this connection further? At some point I'll take a look for myself... –  Pete L. Clark Aug 29 '10 at 3:03
Dear Pete: Can you please give an example of a statement whose proof you consider to be a good illustration of using a "saturated model of the theory of alg. closed fields"? There is nothing about "model theory" in EGA, but my guess is that the underlying mechanism behind what you have in mind must be there in some form. I don't know what your terminology means, which is why I am asking for a representative example. –  BCnrd Aug 29 '10 at 4:42
My impression was the universal domains were a convenient, pre-schemes, way of dealing with generic points by thinking of them as rational points over some larger ideal field. Saturated models are an attempt to do the same thing in more general settings. Often they are a convenience that allows you to avoid the bookkeeping necessary when continually passing to extensions. –  Dave Marker Aug 29 '10 at 14:12
As Pete mentions, the notion of a saturated differentially closed field seems to provide a clear idea of what a universal domain should be for Kolchin's differential algebra. One warning--for some theories,like real closed fields, the existence of saturated models may depend on set theoretic assumptions. In these cases weaker notions may be helpful or one can often argue formally that the assumption is unnecessary for the conclusion. –  Dave Marker Aug 29 '10 at 14:12

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.

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There are plenty of papers of Shimura which use the Weil-Igusa style foundations and also his book on automorphic forms (which contains an appendix "reminding" the reader of Weil stuff). In fact, off the top of my head I can't name a single paper of Shimura in which he uses scheme-theoretic language... –  Pete L. Clark Sep 3 '10 at 14:14
Pete, in 2nd year of grad school I once sat in an interesting course of Shimura's on modular forms in which at one point he was defining "adelization" of some algebraic groups. I was raised my hand and asked "Is that the same as the adelic points of the corresponding group scheme?" He responded "we don't use those words in this course", and that was the end of that. Since Shimura found his viewpoint entirely adequate for what he wanted to do in his research, it seems entirely reasonable that he stuck with what was familiar and sufficient. –  BCnrd Sep 3 '10 at 15:12
Dear Matt: A striking example where seem to lose touch with quasi-projectivity is Weil restriction. By construction of $X = {\rm{Res}}_{R'/R}(X')$ for a finite loc. free $R \rightarrow R'$ and a quasi-proj $R'$-scheme $X'$, unclear if $X$ is quasi-projective; all we see is finite type, septd via val. crit. Even $X' = {\mathbf{P}}^1_ {R'}$ "by hand" is unclear (NB: usually not proper)! For a long time I was mystified by this, so last year I asked Gabber. He gave a very nice proof. It's Prop. A.5.8 in "Pseudo-reductive groups" (even though we didn't need it) since should be more widely known. –  BCnrd Sep 3 '10 at 15:24
Dear Pete, Yes, I know that all Shimura's papers are written in the old language. What I was thinking of was the following foundational paper: "Reduction of algebraic varieties with respect to a discrete valuation of the basis field", in Amer. J. Math. 77 (1955), which provides a framework, in Weil's language, for studying reduction mod p and related notions. –  Emerton Sep 3 '10 at 15:51

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