Some arithmetic terminology: "universal domain", "specialization", "Chow point" - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:17:57Z http://mathoverflow.net/feeds/question/36979 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36979/some-arithmetic-terminology-universal-domain-specialization-chow-point Some arithmetic terminology: "universal domain", "specialization", "Chow point" Qfwfq 2010-08-28T16:17:14Z 2010-09-03T13:38:25Z <p>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.</p> <p>First,</p> <blockquote> <p>What's a "universal domain" (of a given characteristic)?</p> </blockquote> <hr> <p>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}.</p> <blockquote> <p>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?</p> </blockquote> <p>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. </p> <blockquote> <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 $J$"?</p> </blockquote> <p>Also,</p> <blockquote> <p>What's a "specialization over another specializaion"?</p> </blockquote> <hr> <blockquote> <p>What's the "Chow method" to construct the jacobian of a nonsingular curve (of any characteristic)?</p> </blockquote> <hr> <blockquote> <p>What's the "Chow point"? (I suppose it's a concept related to field extensions...)</p> </blockquote> http://mathoverflow.net/questions/36979/some-arithmetic-terminology-universal-domain-specialization-chow-point/36983#36983 Answer by Donu Arapura for Some arithmetic terminology: "universal domain", "specialization", "Chow point" Donu Arapura 2010-08-28T17:10:43Z 2010-08-29T12:47:57Z <p>Dear unkown,</p> <p>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 <a href="http://www.ams.org/notices/199908/fea-raynaud.pdf" rel="nofollow">http://www.ams.org/notices/199908/fea-raynaud.pdf</a> which should help.</p> <p><strong>Addendum</strong> 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.</p> <p>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 $Y=X$.</p> http://mathoverflow.net/questions/36979/some-arithmetic-terminology-universal-domain-specialization-chow-point/37000#37000 Answer by Pete L. Clark for Some arithmetic terminology: "universal domain", "specialization", "Chow point" Pete L. Clark 2010-08-28T21:39:17Z 2010-08-29T17:16:59Z <p>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).</p> <p>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. </p> <p><b>Addendum</b>: To clarify my point in the first paragraph: I am not claiming that scheme theory <em>lacks the power to express</em> 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 <a href="http://math.stanford.edu/~conrad/papers/Kktrace.pdf" rel="nofollow">Brian Conrad's scheme-theoretic exposition of Chow's $K/k$-trace</a>. 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!</p> http://mathoverflow.net/questions/36979/some-arithmetic-terminology-universal-domain-specialization-chow-point/37615#37615 Answer by Emerton for Some arithmetic terminology: "universal domain", "specialization", "Chow point" Emerton 2010-09-03T13:38:25Z 2010-09-03T13:38:25Z <p>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.</p> <p>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.)</p> <p>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.</p>