Properties of the Zariski-Riemann topology on the set of valuations

One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.

It's easy to see that $S_K$ is never Hausdorff. One can show that it's quasi-compact.

I'm wondering if we can say more about $S_K$. Does it have an algebraic structure? Is it a stack?

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The natural ring to put on each open set is the intersection of all the valuation rings. This gives you the correct answer for curves, when you know what the algebraic structure should be. I don't know enough about stacks to figure out if it is one. – Will Sawin Nov 21 '11 at 3:54
As a set or topological space, is the inverse limit of all proper models of the function field. It is probably best to treat it as a formal inverse limit of varieties. – Ben Wieland Nov 21 '11 at 4:59