Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations.

First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of another field $k$. We only consider the case $k=\mathbb{Q}$ or $\mathbb{C}$. By definition, A *model* of $K$ is a variety $V\subset \mathbb{CP}^n$ defined over $k$, such that the rational function field of $V$ over $k$ is isomorphic to $K$. We define the *underlying topological space* of $V$ to be a space, whose points are irreducible subvarieties of $V$, endowed with Zariski topology.

Now comes the interesting thing: Zariski gave an homeomorphism between the space of valuations on $K/k$ and the inverse limit of underlying topological spaces of all models of $K$.

**Question:** Plz give me some concret examples of the above correspondence.

The only example I know is that, given an irreducible hypersurface of a model $V$, one can count the order of rational functions on $V$ over the hypersurface. This gives a discret rank one valuation.

Is there some other easily-described points in the inverse limit, whose corresponding valuations are non-discret, or of higher rank?

Other comments are welcome!

thesmooth model of C. This is the motivating example (no inverse limit needed, and well known before Zariski). First "interesting" case: surfaces. As an example, if p is a point of the curve C on S, a rank 2 valuation of K(S) counts order on C and intersection with C at p. The corresponding point in the Zariski-Riemann space is the limit of all points on C infinitely near to p. References: Zariski-Samuel "Commutative Algebra", Casas-Alvero "Singularities of plane curves". $\endgroup$ – quim Sep 20 '11 at 9:236more comments