Algebraic Geometry
Note on quasi-polarized canonical Calabi–Yau threefolds
Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 415-420.

Let $\left(X,L\right)$ be a quasi-polarized canonical Calabi–Yau threefold. In this note, we show that $|mL|$ is basepoint free for $m\ge 4$. Moreover, if the morphism ${\Phi }_{|4L|}$ is not birational onto its image and ${h}^{0}\left(X,L\right)\ge 2$, then ${L}^{3}=1$. As an application, if $Y$ is an $n$-dimensional Fano manifold such that $-{K}_{Y}=\left(n-3\right)H$ for some ample divisor $H$, then $|mH|$ is basepoint free for $m\ge 4$ and if the morphism ${\Phi }_{|4H|}$ is not birational onto its image, then either $Y$ is a weighted hypersurface of degree $10$ in the weighted projective space $ℙ\left(1,\cdots ,1,2,5\right)$ or ${h}^{0}\left(Y,H\right)=n-2$.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.55
Classification: 14E05,  14J30,  14J32,  14J45
Keywords: birationality, Calabi–Yau threefolds, Fano manifolds, freeness
Jie Liu 1

1 Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
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Jie Liu. Note on quasi-polarized canonical Calabi–Yau threefolds. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 415-420. doi : 10.5802/crmath.55. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.55/

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