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I'm trying to see if it's possible to do an "algebraic geometry 20-questions game"

On an index card there is printed the equation for some algebraic variety $W$, in this case, let's say it's the zero-set of

$x^{7}y^{3} - y^{7}z^{3} + z^{7}x^{3} = 0$.

In the setup of this game, there are three sorts of questions:

  1. Allowed questions: what is the number of rational points on the surface? What are the homology/cohomology/homotopy groups of the surface? In general, these questions are about some property of the algebraic images of $W$. These questions are encouraged in the context of the game. What is the Kodaira dimension of $W$? They do not have to be yes/no questions.

  2. Not allowed questions: "Is some point $(x,y,z,w)$ a part of this surface?" (one could ask this many many times and build up a picture of the surface)

  3. Discouraged questions: "Is $W$ given by the zero-set of $x^{7}+y^{4}z^{8}-xzw^{5}=4$?" (I say discouraged because the point of this exercise is not to brute force an answer, but questions like this are appropriate at the end, when the answer could be yes)


The goal of the game is to determine what $W$ is explicitly (or, more generally, the variety that the asker has in mind), or as Zev puts it

"Is there a finite list of invariants of a variety that determine it completely?"


If such a game is possible, could someone run through a hypothetical transcript of one? (or, to up the level of abstraction: what strategy would you use to play it?)

If such a game is not possible, please explain why not.


EDIT: Clarification: I could have asked "There is an unknown variety $W$: and all that can be determined about it are its invariants, can we tell explicity what sort of variety it is?", but "all that can be determined" is somewhat arbitrary, so I used the frame of a game to provide a reason that there would be limits to the information available about the variety in question.

I'm more interested in the machinery of algebraic geometry that would provide strategies for reducing the number of questions a player would need to ask to determine the variety in question than special cases that reduce to "I'm thinking of a number". In the case of a twenty questions style game: there is an explicit algebraic variety that the one player has in mind, and the other players here need some reasonable strategy for determining what sort of variety it is. (asking a countably infinite number of questions is not an option.)

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    I'm not sure I understand what you mean by "possible." Can you define more precisely the type and number of questions?2011-06-26
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    Were you intending to have the equation be visible only when the cursor points to it, as it currently is, or were you trying to highlight it in some way?2011-06-26
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    @Arturo: yes, the spoiler tag is intentional. In the course of the game I have sketched above, one does not get to see the index card.2011-06-26
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    How is this a math question?2011-06-26
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    @deoxygerbe: Yes, but you aren't asking *us* to guess; if I understand you, you are asking us whether it is possible to have a sequence of questions of the kind you suggest that will, after a suitable number of questions, uniquely identify the variety in the card. And you are asking either for such a sequence, or for a proof that such a sequence will not uniquely identify the variety. *We* aren't playing the game, obviously, so what's the point of (pseudo)-"hiding" the equation here?2011-06-26
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    @Mariano: I think what deoxygerbe is trying to ask is, "Is there a finite list of invariants of a variety that determine it completely?" I doubt that the answer is yes, but I would be interested in seeing explicit examples of non-isomorphic varieties having many invariants / measurements agreeing.2011-06-26
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    @deoxygerbe: Do you restrict yourself to varieties over $\mathbb{C}$?2011-06-26
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    @Zev Chonoles: I do not.2011-06-26
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    Oh well. Someone could edit the question into being more explicit about what the question is! :)2011-06-26
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    In any case, over a sensible field the isomorphism class of a projective variety is a point in a Hibert scheme of finite dimension, so it is determined by finitely many scalars (and the Hilbert scheme itself to which the point belong is determined by finitely many invariants of the variety)2011-06-26
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    @Mariano: With knowledge of the concept of [20 questions](http://en.wikipedia.org/wiki/20_questions), I think the question as posed is perfectly understandable, though algebraic geometry is not a strength of mine.2011-06-26
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    Well, I know about the Hilbert scheme but had never in my life heard about a game called *20 questions*: if the idea is to attract answers, a slightly less colourful description of the point of the question might help :)2011-06-26
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    The more I think about the question, the less sure I am that I understand what the OP is asking, or means to ask. More clarification would certainly help.2011-06-26
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    @deoxygerbe: I still don't really understand what you are asking. Further questions: 1) Are you studying varieties up to isomorphism? If not, then what? 2) What exactly is an "invariant"? Why isn't the isomorphism class an "invariant"? (Note for instance that the **j-invariant** of an elliptic curve is precisely what is needed to classify it up to isomorphism over an algebraically closed field.) 3) What do you mean by determining "what sort of variety" it is?2011-06-26
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    Okay, so here is one possible precisification of the question: is there a finite list of computable invariants of, say, a projective variety over $\overline{\mathbb{Q}}$ that determines it up to isomorphism (over $\overline{\mathbb{Q}}$)? (I say "one possible" because the OP seems to be interested in things like fundamental groups and these are not necessarily computable in the sense that it is not _a priori_ decidable whether two varieties have the same fundamental group...)2011-06-26
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    @Qiaochu: isn't that question equivalent to asking whether there is an algorithm for deciding whether two such varieties are isomorphic? (If so, good question: pretty sure it's open and that many believe the answer will be "no".)2011-06-26
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    @Pete: I guess I can believe that. Is Hilbert's 10th problem open over $\overline{\mathbb{Q}}$?2011-06-26
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    @Qiaochu: Hmm, no, H10 is almost trivially true over any algebraically closed field: it amounts to an effective Nullstellensatz. I see where you're going with this, but the automorphism group of even a projective algebraic variety is only a "locally algebraic group" c.f. http://mathoverflow.net/questions/8812, so a solution to H10 does not necessarily give us a procedure for telling whether the $\overline{Q}$-scheme $\operatorname{Iso}(V_1,V_2)$ has a $\overline{Q}$-point. Still this makes me wonder whether I was thinking about $\mathbb{Q}$-varieties instead...2011-06-26
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    Wait, no, I was remembering correctly! This was the end of Bjorn Poonen's Cantrell Lecture series at UGA a few years back: see http://www-math.mit.edu/~poonen/slides/cantrell3.pdf. I even thought about this problem a bit and wanted to use etale fundamental groups to reduce it to the (known to be undecidable) isomorphism problem for finitely presented groups...but I didn't succeed.2011-06-26
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    Ha, whoops. That actually wasn't the direction I was heading, but it's a better motivation for the question than the one I had, which was pretty nebulous...2011-06-26

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