# 2024-07-22

## Valuations

Let $k=\CC$ and $X$ be a proper variety over $k$.
Write $k(X)$ for its function field.
Recall that a valuation is a group morphism $v: k(X)\units\to (\RR, +)$ where 

- $v(fg) = v(f) + v(g)$
- $v(f+g) \geq \max\ts{v(f), v(g)}$
- $v(k) = 0$

We generalize this slightly to *quasi-monomial valuations*. 
For $X$ of dimension $n$, let $(Y, E)$ be a (log smooth) minimal resolution.
Write $E = \sum E_i$, and e.g. $p = \Intersect_i E_i$, and write $\widehat{\OO_{Y, p}} = k\fps{y_1, \cdots y_r}$.
Then any $f\in \OO_{Y, p}$ can be written as $f= \sum_{\alpha} f_ \alpha y^{ \alpha}$.
A quasi-monomial valuation $v_m$ is given by $v_m(f) = \min\ts{ m\cdot \alpha }$ ranging over \( \alpha \).
Write $QM(Y, E)$ for the set of quasi-monomials depending on the choice of $(Y, E)$.

For \( Y \mapsvia{\varphi} X \) a resolution with exceptional divisor $E$, write $A_X(E) = \ord_E(K_{Y/X}) + 1$ for the log discrepancy.
For $v_ \beta \in QM_p(Y, E)$ with $p = \Intersect E_i$, define $A_X(v_ \beta) \da \sum \beta_i A_X(E_i)$.