I am trying to develop a good geometric intuition and to understand the motivation behind Kawamata's definition of a relative movable Cartier divisor in Section 2 of reference [1]: 


[1] Y. Kawamata, Crepant blowing-ups of three-dimensional canonical singularities and its application to degenerations of surfaces; Ann. of Math. 127 (1988), 93-163; [JSTOR](https://www.jstor.org/stable/1971417).


**Definition.** Let $f:X\to S$ be a surjective projective morphism between normal  

A Weil divisor $D$ is said to be $f$-movable if 


$$
f\_\* O\_X(D)\neq 0\quad \ \text{and}\ \quad codim\ Supp\ Coker\Big(\ f^\* f\_\* O\_X(D)\to O\_X(D)\Big)\geq 2. 
$$


**Questions.**


(1) What is the geometric intuition and the motivation behind this definition?


(2) What are the basic examples and nonexamples?



Intuition behind Kawamata's definition of a relative movable Cartier divisor
The base locus of a divisor $D$ on $X$ is the same as those points where $\mathscr O\_X(D)$ is **not** generated by global sections, which can be identified with the locus where the natural map 
$$
\tag{$\star$}
H^0(X,\mathscr O\_X(D))\otimes \mathscr O\_X \to \mathscr O\_X(D) 
$$
is **not** surjective. 


The relative version of global sections is the sheaf $f\_\*\mathscr O\_X(D)$. (If $f$ maps to a point, or even to an affine scheme, it is literally just that). So, the relative version of $(\star)$ is the map
$$
\tag{$\star\star$}
f^\*f\_\*\mathscr O\_X(D)\to \mathscr O\_X(D). 
$$
(Notice that $H^0$ is replaced by $f\_\*$ and $\\_ \otimes \mathscr O\_X$ by $f^\*$).
Finally, this means that the relative version of the base locus is the locus where the map in $(\star\star)$ is **not** surjective, i.e., the support of the cokernel of that map. 


