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.)