I am interested in valuations on a function field $K=k(X)$ of some say smooth, projective $k$-variety $X$ of dimension $n$, where $k$ is some (algebraically closed) field (that implies trdeg$(K/k) = n$). These valuations shall also be trivial on $k$. In particular valuations that come from a flag $$ X_{\bullet}=\{\{p\} = X_0 \subset X_1 \subset \dots \subset X_n = X\} $$ of closed subvarieties with codim$(X_i) = i$, such that $X_{i+1}$ is a Cartier divisor of $X_i$ in a neighborhood of $p$. The construction can be found for example in http://www.math.jussieu.fr/~boucksom/publis/Okounkov_Bourbaki.pdf Exemple 2.17. Every such valuation has value group $\mathbb{Z}^n$ and is of maximal rational rank (i. e. of rational rank = $n$). I am wondering if any valuation on $K$ of maximal rational rank can be realized as a flag valuation on some (or any say smooth projective) $k$-variety with function field $K$. What do you think?
Add a comment
|