The following comes(paraphrased) from "Basic Algebraic Geometry I" by Shafarevich.
A representative of a birational equivalence class of quasiprojective varieties is called a model.
Let $ k $ be an algebraically closed field, and $ X $ be an $ n-$dimensional non-singular quasiprojective model. We say that $ X $ is universal if it contains all the local rings of $ K = k (X) $ satisfying:
(1) $ \mathcal{O} $ is a subring of $ K $ with $ k \subsetneqq \mathcal{O} \subsetneqq K; $
(2) $ \mathcal{O} $ is a local ring, and its maximal ideal $ \mathfrak{m} = (u_{1},\dots,u_{n}); $
(3) $ K $ equals the field of fractions of $ \mathcal{O}. $
One might ask for the existence, in each birational equivalence class, of a model $ X $ that would be universal in the sense that the local rings $ \mathcal{O}_{x} $ of points $ x \in X $ exhaust all the local subrings of the field $ K = k(X) $ that satisfy conditions (1),(2) and (3). However, no such model can exist, for the same reasons. Namely if $ \sigma : X' \rightarrow X $ is the blowup of $ X $ with centre in $ \xi, $ then the local rings of points $y \in \sigma^{-1}(\xi) $ are not equal to any of the local rings $ \mathcal{O}_{x} $ with $ x \in X. $
How can I do this verify that this is the case?
Write $z$ for some point in $\xi$ and $y_1,y_2$ for two points in $\sigma^{-1}(\xi)$. It can't be the case that the local ring of $y_i$ could be the same as a local ring at $x$ chosen from outside $\xi$: the maximal ideal at $y_i$ contains all the functions which vanish on $\xi$, for instance, but this is not the case for any point $x\notin\xi$. So if the local ring of $y_i$ is the same as a local ring of some point $x\in X$, we get that $x\in \xi$.
Conversely, any function vanishing at $z$ must vanish at $y_1$ and $y_2$. But the maximal ideal at the valuation ring corresponding to $y_1$ contains a function which vanishes at $y_1$ but not $y_2$. Therefore the maximal ideal of $\mathcal{O}_{y_1}$ can't be the same as any maximal ideal of any valuation ring corresponding to a point $x\in X$.