19.3.1 The algebraic setting07F4BAAD0C8E

Given any admissible flag $Y_{\bullet }$, we can define a rank $n$ valuation $\nu _{Y_{\bullet }}\colon \mathbb {C}(X)^{\times }\rightarrow \mathbb {Z}^ n$. Here we consider $\mathbb {Z}^ n$ as a totally ordered Abelian group with the lexicographic order. We sometimes write $\mathbb {Z}^ n_{\mathrm{lex}}$ to emphasize this point.

If we identify the elements in $\mathbb {Z}^ n$ with a row vector, the automorphism group $\operatorname{Aut}(\mathbb {Z}^ n_{\mathrm{lex}})$ of $\mathbb {Z}^ n_{\mathrm{lex}}$ is then identified with the subgroup of $\operatorname{GL}(n,\mathbb {Z})$ consisting of matrices of the form $\mathrm{I}+U$, where $\mathrm{I}$ is the identity matrix and $U$ is a strictly upper triangular matrix with elements in $\mathbb {Z}$.

We recall the definition of $\nu _{Y_\bullet }$. Let $s^{(0)}=s\in \mathbb C(X)^\times $. For $i=1,\ldots ,n$, after localizing around $Y_ n$, choose a local defining equation $t_ i$ of $Y_ i$ in $Y_{i-1}$, and set

Then

It is easy to verify that $\nu _{Y_{\bullet }}$ is indeed a rank $n$ valuation.

The same construction applies to a non-zero section $s\in \mathrm H^0(X,L)$ after choosing a local trivialization of $L$ near $Y_ n$; the result is independent of this choice, since units have valuation $0$. We also write $\nu _{Y_\bullet }(D)$ for a non-zero effective Cartier divisor $D$, and extend the notation $\mathbb Q$-linearly to non-zero effective $\mathbb Q$-Cartier divisors. We also set $\nu _{Y_{\bullet }}(0)=0$.