What does $|mK_V|=|M|+F$ mean for linear system $|mK_V|,|M|$?

Question

I apologize for the ambiguity of the title, the question comes from M.Reid's Young persons guide to canonical singularities (p.355), and I saw similar constructions in many other places. In fact this question is more than explanations of notations.

Let $V$ be a smooth projective variety of general type. $K_V$ is a canonical divisor of $V$. Reid wrote:

Let $V' \to V$ be a resolution of the base locus of $|mK_V|$; in view of the birational invariance of $H^0(mK_V)$, I can replace $V$ by $V'$, so assume that $$|mK_V|=|M|+F,$$ where $|M|$ is a free linear system and $F$ the fixed part.

I have three questions:

(1) What does $|mK_V|=|M|+F$ mean?

I understand it in the following way: Let $F'$ be the base locus of $|mK_V|$, then $|mK_V|$ defines a morphism $$\phi: V - F' \to \mathbb{P}^N,$$ with the closure of image $\overline{\phi(V-F')}=X$ and let $V'$ be the desingulization of the graph of $\phi$. Thus we have $$V \xleftarrow{q} V' \xrightarrow{p} X.$$

Let $M:=p^*(\mathcal{O}_X(1))$, and $F:=p^{-1}(F')$, then one should have $$M|_{(V'-F)} \cong mK_{V'}|_{(V'-F)}.$$ However, I don't know how to make sense of $|mK_V|=|M|+F$.

(2) Why $F$ is a (Weil) divisor? (Moreover, I remember in somewhere it is claimed to be a Cartier divisor). It seems very strange to me that the fixed locus of linear system is a divisor, could it have higher codimension?

(3) It seems Reid got $$mK_V \cong \phi^*\mathcal{O}_X(1) + F$$ from above relation, why this is the case?

Answer 1

To answer the questions in a different order:

2) You're right that in general the base locus of a linear system could be higher-codimensional; it's not hard to come up with examples. The point here is that Reid has written: "Let V′→V be a resolution of the base locus of $|mK_V|$... I can replace V by V′...". That is, he has blown up any components of the base locus that aren't divisors. Then since $K_{V'} = \pi^* K_V + aE$ for some nonnegative $a$ (assuming $V$ is canonical), the base locus of $|mK_{V'}|$ will be a divisor.

1) One way to say this is that every effective divisor in the linear system $|mK_V|$ can be written uniquely in the form $D+F$, where $D \in |mK_V|$. Another way to say it is that we have an equality of line bundles $mK_V = M \otimes O_V(F)$; since the latter bundle has a unique global section (this is exactly what fixed part means) we get an isomorphism between sections of $mK_V$ and those of $M$, by tensoring with the unique section of $O_V(F)$.

3) Whenever you have a free linear system $|M|$ it defines a map $\phi$ to projective space such that $\phi^* O(1) = M$. Now just plug this into the expression $mK_V = M \otimes O_V(F)$, and (confusingly) switch to additive notation!