Consider something like baby Grothendieck motives, by which I'll mean classes of algebraic varieties X modulo relations [X] - [Y] = [X\Y], [X x Y] = [X] x [Y], [A] * [A^-1] = [point] where A is an affine line and A^-1 the formal inverse. Also, the ring will be complex numbers. If you want, you can also take idempotent completion.

Now those things can be added and multiplied, so they form a ring, call it M. And for things that form a ring you can study their Spec. E.g. they have points.

Question: what are the scheme properties of Spec M? How to describe its points?

For example, one point is Euler characteristics \chi \in \Spec M, since it's additive and multiplicative. I think there's also a plane there given by Hodge numbers h^{p,q}, since Hodge polynomial at a given point satisfies those relations too. Are there any other points?

Any more information?