I feel the following fact has been used in many argument in algebraic geometry, but I was not be able to prove it or find the precise reference:


Let $X$ be a $\mathbb{Q}$-factorial variety with log canonical singularities, and $Z \subseteq X$ be a subvariety. Then the set $$\{E \mid E {\rm{~is~ an~ exceptional~ divisor~ of~ some~ resolution~ }}Y \to X, {\rm ~such ~ that~} f(E)=Z {\rm~and~} discrepancy(K\_X, E) \leq 1 \}$$ is finite.


Any suggestion for the proof or references is welcome!!



Finiteness of geometric valuations
Ok, let me show the the following which is well known to experts (although I'm not sure where the right reference is):


**Proposition:** *If $(X, \Delta)$ is KLT (or PLT), then there are at most finitely many divisorial valuations corresponding to divisors $E$ on birational models of $X$ such that the discrepancy of $K\_X + \Delta$ along $E$ is $\leq 0$*.


(Recall that if $X' \to X$ is birational then the coefficient of $K\_{X'} - \pi^\*(K\_X + \Delta)$ along a prime divisor $E$ is called the discrepancy of $K\_X + \Delta$ along $E$).


**Proof:** First let $\pi : Y \to X$ be a log resolution of $(X, \Delta)$. Consider then the simple normal crossings pair $(Y, -K\_Y + \pi^\*(K\_X + \Delta))$. This pair is probably non-effective, but it is SNC. 


Let $Y' \to Y$ be the composition of a sequence of blowups along the strata of $(Y, -K\_Y + \pi^\*(K\_X + \Delta))$ such that in $(Y', -K\_{Y'} + \pi'^\*(K\_X + \Delta) )$ every divisor on $Y'$ with discrepancy $\leq 0$ is disjoint from every other such divisor. 


(This is not so hard to see how to do, note if I have two divisors with discrepancy $-1 < a \leq 0$ that intersect, then the blowup of their intersection separates the two divisors and creates a new divisor with discrepancy strictly bigger than that of the two original divisors. Keep doing this until separation is achieved). 


At this point, we have reduced to the case of a pair $(Y', \Delta' = -K\_{Y'} + \pi'^\*(K\_X + \Delta) )$ where $Y'$ is smooth, $\Delta'$ is SNC (not necessarily effective) and $\Delta'$ has finitely many prime divisors with coefficient $\geq 0$, all of which are disjoint from each other. In fact, it is easy to see that any exceptional divisor over $(Y', \Delta')$ has discrepancy $> 0$. Hence we are done. $\blacksquare$


**Remark:** This is false for log canonical singularities as mentioned in the comments (ie, consider $(\mathbb{A}^2, \text{Div}(xy))$).


