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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? –  Qiaochu Yuan Jun 26 '11 at 3:08
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How is this a math question? –  Mariano Suárez-Alvarez Jun 26 '11 at 3:13
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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. –  Zev Chonoles Jun 26 '11 at 3:16
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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) –  Mariano Suárez-Alvarez Jun 26 '11 at 3:24
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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? –  Pete L. Clark Jun 26 '11 at 13:11
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1 Answer 1

I'm not sure I'm grokking the spirit of this game (is it actually meant to be played, or is this some kind of thought experiement? if the latter, to what purpose?), but:

Suppose that one restricts attention to algebraic varieties that can be specified precisely using a finite amount of data: for instance algebraic varieties over $\overline{\mathbb{Q}}$. The number of isomorphism classes of such varieties is countably infinite. For instance, I can take an elliptic curve with $j$-invariant any given algebraic number. If I want them to be nonisomorphic even as abstract schemes rather than schemes over $\overline{\mathbb{Q}}$ then I can take an elliptic curve with complex multiplication by the maximal order of $\mathbb{Q}(\sqrt{-d})$ as $d$ ranges over all squarefree positive integers.

If there are infinitely many possible answers, then any finite sequence of questions will not suffice to determine the isomorphism class of the variety. Conversely, if you allow infinitely many questions, then you can enumerate the isomorphism classes of varieties over $\overline{\mathbb{Q}}$ and ask as your $n$th question: is your variety isomorphic to $V_n$ (the $n$th variety in my list)? You say you want to "discourage" such questions, but I don't really see what the point of that is: of course you could ask a more complicated infinite sequence of questions, but why?

Added: the above construction requires some knowledge of arithmetic geometry. This is not really necessary, so let me give a simpler one: suppose that I let the questioner in advance know that I am picking some Fermat curve

$F_n: x^n + y^n - z^n = 0$

for $n \in \mathbb{Z}^+$. Then the genus of $F_n$ is the increasing function $\frac{(n-1)(n-2)}{2}$, so these are certainly nonsisomorphic varieties. Then it is clear that we are really playing the game "I am thinking of a positive integer" and $20$ questions -- or any predetermined finite number of questions -- will not suffice. However, if the questioner is allowed to ask as large a finite number of questions as she needs, then of course eventually she will be able to guess the number. It is the same for isomorphism classes of varieties over $\overline{\mathbb{Q}}$ since this forms a countably infinite set. Or at least it is the same ignoring issues of algorithmic effectivity: I am assuming that whatever question gets asked gets a yes/no answer. If effectivity is actually in question, I recommend that we try again with a different formulation of the question.

Let me also say that the OP didn't mention anything about "up to isomorphism" and indeed questions like "Is this point on the variety?" don't make sense in the context of varieties up to isomorphism. (Anyway, "is this point on the variety?" is a silly question to ask even of, say, algebraic subsets of affine $n$-space since in that in no case is it possible to determine what the subset is by asking any finite number of such questions.) If we are talking about abstract varieties not up to isomophism, for silly reason this forms a proper class so is highly inappropriate for playing $20$ questions (or even $\kappa$ questions, for any cardinal number $\kappa$). If we mean, say, projective varieties given as closed subsets of projective space over $\mathbb{P}^n_{\overline{\mathbb{Q}}}$ then it certainly is algorithmically effective to ask "Are you thinking of the subset of $\mathbb{P}^n$ given by the vanishing of these polynomials?" so the question manifestly collapses to "I'm thinking of a number..."

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Okay, but unlike other possible invariants, it's not a priori obvious that the invariant "is my variety isomorphic to $V_n$" is computable or efficiently computable. (Actually, is it? I have no idea.) –  Qiaochu Yuan Jun 26 '11 at 4:18
    
@Qiaochu: you're right, it's not obvious that it's computable. In fact, IIRC this is an open question. I guess I was just playing by the rules of the game: we are allowed to ask yes/no questions and assume that we will get answers... –  Pete L. Clark Jun 26 '11 at 5:39
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@Rasmus: No, I didn't: I was assuming that the answer to each question has to be "yes" or "no", as in the usual game of $20$ questions. (I have to say that this is yet another way in which the framing of the question is rather poor.) As you point out, the OP does not in fact formulate it this way, but as I point out in my latest comment to the question, I don't really understand that formulation. In its more general version, the question depends so critically on what is an allowable "invariant" that it seems impossible to answer in its present form. –  Pete L. Clark Jun 26 '11 at 13:42
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@Rasmus: Believe it or not, I am trying not to pick on the OP too much, so I tried to simply ignore some parts of the phrasing that don't seem quite correct or coherent. This caused me to ignore most of the "allowed questions" part, since: a) the OP first says "variety" and then says "surface"; b) the OP says "rational points" but this doesn't make sense without specifying some ground fields; c) the OP talks about things like homotopy groups which a priori make sense only for varieties over $\mathbb{C}$ but in response to a question the OP says he is not assuming this... –  Pete L. Clark Jun 26 '11 at 13:50
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...(If the variety is over something like $\overline{\mathbb{Q}}$ then the isomorphism class of the fundamental group in general depends on the embedding into $\mathbb{C}$!), the OP says "In general, these questions are about some property of the algebraic images of W" but this is not true for the topological invariants (except insofar as they are properties of $W$, which is an algebraic image of $W$), and so forth. So some interpretation / adjustment of the question seems to be required, at present. –  Pete L. Clark Jun 26 '11 at 13:55
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