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

Let (X,L) be a quasi-polarized canonical Calabi–Yau threefold. In this note, we show that |mL| is basepoint free for m4. Moreover, if the morphism Φ |4L| is not birational onto its image and h 0 (X,L)2, then L 3 =1. As an application, if Y is an n-dimensional Fano manifold such that -K Y =(n-3)H for some ample divisor H, then |mH| is basepoint free for m4 and if the morphism Φ |4H| is not birational onto its image, then either Y is a weighted hypersurface of degree 10 in the weighted projective space (1,,1,2,5) or h 0 (Y,H)=n-2.

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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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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