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arXiv 1809.00481math.AG
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关于准极化正则Calabi-Yau三重性的注记

Note on quasi-polarized canonical Calabi-Yau threefolds

Jie Liu

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摘要：

设$(X,L)$是一个准极化正则卡-丘三重图。本文证明了对于$m\geq 4$，$\vert mL\vert$是无基点的.此外，如果态射$Φ_{\vert 4L\vert}$在其像和$h^0(X,L)\geq 2$上不是双有理的，则$L^3=1$。作为一个应用，如果$Y$是一个$n$维Fano流形，使得$-K_Y=(n-3)H$对于某个充分因子$H$，则$\vert mH\vert$对于$m\geq 4$是基点自由的，并且如果态射$Φ_{\vert 4H\vert}$在其像上不是双有理的，则$Y$是加权射影空间$\mathbb{P}(1,\cdots,1,2,5)$或$h^0(Y,H)=n-2$中的次数为$10$的加权超曲面.

英文摘要：

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

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