It is proved in this paper by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the $\mathcal{O}_X$-algebra $$\mathcal{R}_X(D):=\oplus_{m\geq 0}\mathcal{O}_{X}(mD)$$ is finitely generated.

I want to know:

(1) If the above result still true for higher dimensional varieties(with mild singualrities or even a variety of Fano type)?

(2) My interest in this problem comes from using its "Corollary" in higher dimensional case (see Corollary 4.5 loc.cit):

Let $X$ be a 3-dimensional variety with terminal singularities, then there exists a projective birational morphism $\phi: Y \to X$ such that: (1) $Y$ has at most $\mathbb{Q}$-factorial terminal singularities, and (2) $\phi$ is small.

I want to know if in the higher dimensional case, there still exists such small modification of a variety to be a $\mathbb{Q}$-factorial variety?


Indeed, this is fine. In fact it is an exercise in Kollár's arXiv notes (of course utilizing the MMP):


see in particular exercises 90-110.

The upshot is that everything is fine for KLT ambient spaces.

  • $\begingroup$ Dear Karl Schwede, thank you very much!! $\endgroup$ – Li Yutong Dec 2 '14 at 14:32

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