Let $(\sM,\sL)$ be a smooth $(n+1)$-dimensional variety polarized by an ample and spanned line bundle $\sL$. Let $A$ be a smooth member of $|\sL|$. Assume that $n\geq 4$ and that $(A,H_A)$ is a Mukai variety, i.e., $-K_A\approx (n-2)H_A$ for some ample line bundle $H_A$ on $A$. Let $H$ be the line bundle on $\sM$ which extends $H_A$. We show that $\sM$ is a Fano variety and either $H$ is ample, in which case the cones of effective $1$-cycles $NE(A)$ and $NE(\sM)$ on $A$ and $\sM$ coincide, or $\grk(H)=n$, $H$ is semiample and $(\sM,\sL)$ has a structure of a conic fibration. Then most of the paper is devoted to classify the pair $(\sM,\sL)$ in the case when $\sL$ is very ample.
Mukai varieties as hyperplane sections
BELTRAMETTI, MAURO CARLO;
2004-01-01
Abstract
Let $(\sM,\sL)$ be a smooth $(n+1)$-dimensional variety polarized by an ample and spanned line bundle $\sL$. Let $A$ be a smooth member of $|\sL|$. Assume that $n\geq 4$ and that $(A,H_A)$ is a Mukai variety, i.e., $-K_A\approx (n-2)H_A$ for some ample line bundle $H_A$ on $A$. Let $H$ be the line bundle on $\sM$ which extends $H_A$. We show that $\sM$ is a Fano variety and either $H$ is ample, in which case the cones of effective $1$-cycles $NE(A)$ and $NE(\sM)$ on $A$ and $\sM$ coincide, or $\grk(H)=n$, $H$ is semiample and $(\sM,\sL)$ has a structure of a conic fibration. Then most of the paper is devoted to classify the pair $(\sM,\sL)$ in the case when $\sL$ is very ample.
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