The proof of Lemma 5-1-5 in this 1987 paper by Kawamata, Matsuda and Matsuki (link on Projecteuclid) seems to say that a variety with terminal singularities is $\mathbb{Q}$-factorial ( I only need the case $ \Delta =0$ in which case, weak log terminal might be the same as just terminal.). Does anyone know how to prove this, or a place where this is proved?
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3$\begingroup$ In dimension two all klt singularities are $\mathbb{Q}$-factorial. In dimension three there are singularities which are terminal and not $\mathbb{Q}$-factorial, for instance the cone over $\mathbb{P}^1\times\mathbb{P}^1$. The Lemma you cite only talks about fibrations, I don't know how it would imply such statement. In any case, if you have a klt singularity, you can make it $\mathbb{Q}$-factorial with a small morphism, this is called a small $\mathbb{Q}$-factorialization. $\endgroup$– Joaquín MoragaCommented Feb 5, 2020 at 21:34
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1$\begingroup$ I believe the paper assumes at the very beginning of section 5-1 that $X$ is $\mathbb Q$-factorial? As @JoaquínMoraga says, this is not true otherwise. $\endgroup$– Devlin MalloryCommented Feb 6, 2020 at 3:39
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I do not know how you get this conclusion from the paper of Kawamata, Matsuda and Matsuki. But this is not true in dimension at least $3$. Of course in dimension 2 every terminal singularity is smooth.
The easiest example of a non-$\mathbb{Q}$-factorial terminal singularity in dimension $3$ is the hypersurface $(xy=zw)\subset \mathbb{A}^4_{x,y,z,w}$. It is terminal as it admits a crepant small resolution. On the other hand, it is not $\mathbb{Q}$-factorial, as there are $2$ Weil divisors $D=(x=z=0)$ and $D'=(y=w=0)$ such that the intersection $D\cap D'$ is exactly one point, the origin. This contradicts the fact that the intersection of $2$ $\mathbb{Q}$-Cartier divisor has codimension at least $2$.
