3
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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

http://arxiv.org/pdf/0809.2579.pdf

see in particular exercises 90-110.

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.