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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    $\begingroup$ 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 Moraga Feb 5 at 21:34
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    $\begingroup$ 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 Mallory Feb 6 at 3:39

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