I'm starting to read about valuations.
Given a real valuation $v$ on a (projective complex) variety $X$, centered at $x$, and a line bundle $L$, we can define $v(s)$ for $s \in H^0(X,L)$ by considering a local trivialization of $L$ near $x$, and evaluating at the regular function corresponding to the restriction of $s$.
My questions are:
1) Why is this independent of trivialization?
2) Is there always $s \in H^0(X,L)$ such that $v(s) = 0$?
For (1)
Suppose we have a trivialization in a neighborhood $U$ of the centre. This amounts to saying the restriction of the sheaf is generated by a section $s_0$ on $U$, and any other section $s$ is of the form $fs_0$ for some regular $f$ on $U$. We would then evaluate $v(s) = v(f)$ . Now suppose we choose a different section to generate the sheaf, $t_0 = g s_0$.
According to the new trivialization $v(s) = v(fg) = v(f) + v(g)$
But $g$ is a section of $ \mathcal{O}_U^\times $, in particular $g$ and $g^{-1}$ are regular in a neighborhood of the center $c_X(v)$. Therefore $v(g) = 0$ and the result is independent of trivialization.