Intuition behind Kawamata's definition of a relative movable Cartier divisor

Question

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.

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?

Answer 1

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.

Answer 2

Let me try to say something that might be useful.

At the risk of stating the obvious, the motivation is to extend the notion of movable divisor (class) to the relative setting. If you haven't already, you should check that when $S$ is a point, you recover the usual definition of movable divisor class --- namely, a class whose base locus has codimension at least 2.

In the absolute case, the base locus consists of those points that global sections of $O_X(D)$ can't avoid. In the relative case, the support of that cokernel sheaf consists of points that sections of $O_X(D)$ can't avoid, even after we just restrict to the preimage of an open set on $S$.

This notion is important for the following reason. In the non-relative case, if $L$ is a movable divisor class on $X$, then we get a contracting rational map $$ X \dashrightarrow \operatorname{Proj} \oplus_{m \geq 0} H^0(X,mL)$$ and in fact all contracting rational maps from $X$ arise in this way.

Now in the relative case, say we have a morphism $f:X \to S$ and we want to understand contracting rational maps over $S$: in other words, varieties $X'$ with a morphism $f':X \to S$ and a contracting map $\varphi: X \dashrightarrow X'$ such that $f' \circ \varphi = f$.

Then we need to consider $f$-movable divisors on $X$: if $L$ is an $f$-movable divisor class on $X$ then we get a contracting rational map over $S$ by taking the relative proj: $$ X \dashrightarrow \operatorname{Proj}_S \oplus_{m \geq 0} \,f_*(mL)$$ and again all contracting rational maps over $S$ arise in this way.

One application is the following. Suppose $Y$ is a smooth 3-fold of general type and we want to understand its minimal models. Let $X$ be a minimal model of $Y$: then it has a canonical model $f: X \to X_{can}$. Now if $X'$ is any other minimal model, then there is a rational map $\varphi: X \dashrightarrow X'$ which is an isomorphism in codimension 1 and a morphism $f': X' \to X_{can}$ such that $f = f' \circ \varphi$. So by the above, all minimal models of $X'$ (equivalently, of $Y$) come from $f$-movable divisors on $X$.

Kawamata–Matsuki used this to prove that 3-folds of general type have only finitely many minimal models. Their (very short!) proof is based on analysing the cone of $f$-movable divisors.

Later, Kawamata generalised this to prove that if $Y$ is any 3-fold of positive Kodaira dimension, it has only finitely many minimal models up to isomorphism.