Algebraic geometry "20 questions" 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:


*

*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.

*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)

*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.)
 A: 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..."  
