# Are terminal singularities $\mathbb{Q}$-factorial?

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?

• 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. – Joaquín Moraga Feb 5 at 21:34
• 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. – Devlin Mallory Feb 6 at 3:39