Let $L$ be a pseudo-effective divisor, we may define its numerical fixed part $N\_{\sigma}(L)$. How to prove it is a divisor? I know there is a proof in Nakayama's book, but I can't find this book.



How to prove the existence of divisorial Zariski decomposition?
By definition, you first define $\sigma\_\Gamma(L)$ for big divisors and then you take the limit.


In other words, if $L$ is big, then clearly $\sigma\_\Gamma(L)$ is non zero for only finitely many divisors. Indeed, $L=A+B$ with $A$ ample $\mathbb Q$-divisor, and $B\ge 0$. Thus $\sigma\_\Gamma(L)>0$ implies $\Gamma$ is contained in the support of $B$. 


If $L$ is pseudo-effective, then you define 
$$\sigma\_\Gamma(L)=\lim\_{\epsilon\to 0^+} \sigma\_\Gamma(L+\epsilon A)$$
where $A$ is ample (it is easy to check that the definition does not depend on the choice of $A$). 


It follows that if $\sigma\_\Gamma(D)=\alpha>0$ then $D-\alpha\Gamma$ is pseudo-effective. Finite generation of $N^1(X)$ implies that there can only be finitely many $\Gamma$ with such a property. 


