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!



The space of valuations of a function field
Simply put (more or less already said above). Let $K=C(x,y)$ and let $xi(t)$ be a generalized power series (a power series with well-ordered exponents) where the exponents are non-negative real numbers. Assume (this is important) that $P(t,\xi(t))\neq 0$ for any $P\in K$. Then the map: $\nu(P(x,y))=ord\_{t}(t,\xi(t))$ is a valuation (the "order of contact of P with $(t,\xi(t))$").


Notice that if $\xi(t)=t^{\pi}$, for example, the valuation has rational rank $2$. If $\xi(t)$ corresponds to an analytic branch of a curve (non-algebraic), the valuation has rational rank $1$, etc.


By the way, [here](http://books.google.es/books?id=su_4FI26xbkC&pg=PA104&lpg=PA104&dq=valuation+projective+limit+favre&source=bl&ots=ySTFyYb9wa&sig=AaEjpnVGw-n7lE_sf3BkShPQmaI&hl=en&ei=k6N4To3gDYex8QO6_qTBDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBcQ6AEwAA#v=onepage&q&f=false) you may find something useful. That book may be of help.


You should read something about point blowing-ups and then you understand the projective limit thing. But without it, it gets somewhat too algebraic.


