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!
Best Answer
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 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.