Note on quasi-polarized canonical Calabi-Yau threefolds

Abstract

Let $(X,L)$ 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 $h^0(X,L)\ge 2$, 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 $|mH|$ is basepoint free for $m\ge 4$ and if the morphism ${\Phi }_{|4H|}$ 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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