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?