The Zariski-Riemann Space as a Universal Model for the Birational Geometry of a Function Field

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf on a normal model, we show how to extend it to a sheaf on $\ZR (K/k)$ and we prove that both operations are compatible with pushforwards and pullbacks. As a result, we study the cohomology theory of the Zariski-Riemann space and we show that, on each local model, it yields the usual sheaf cohomology. Lastly, we focus on ideal-sheaves and we interpret blow-ups using our framework