Let $(\sM,\sL)$ be a smooth $4$-dimensional variety polarized by a very ample line bundle $\sL$. Let $\sA$ be a smooth member of $|\sL|$. Assume that $\sA$ is a Fano threefold of index one, with $-K_\sA\cong \sH_\sA$ for some ample line bundle $\sH_\sA$ on $\sA$. Let $\sH$ be the line bundle on $\sM$ which extends $\sH_\sA$. Up to sporadic examples and four special classes of pairs $(\sM,\sL)$, and up to taking the first reduction $(M,L)$ of $(\sM,\sL)$, we show that $M$ is a Fano variety, and the cones of effective $1$-cycles $NE(A)$ and $NE(M)$ coincide, where $A$ is the image of $\sA$ under the first reduction map. We also show that there exists a new polarization $H$ on $M$ and our main result is proving that the usual adjunction process on $(M,H)$ terminates, this leading to a coarse classification of $(\sM,\sL)$.
Fano threefolds as hyperplane sections
BELTRAMETTI, MAURO CARLO;
2005-01-01
Abstract
Let $(\sM,\sL)$ be a smooth $4$-dimensional variety polarized by a very ample line bundle $\sL$. Let $\sA$ be a smooth member of $|\sL|$. Assume that $\sA$ is a Fano threefold of index one, with $-K_\sA\cong \sH_\sA$ for some ample line bundle $\sH_\sA$ on $\sA$. Let $\sH$ be the line bundle on $\sM$ which extends $\sH_\sA$. Up to sporadic examples and four special classes of pairs $(\sM,\sL)$, and up to taking the first reduction $(M,L)$ of $(\sM,\sL)$, we show that $M$ is a Fano variety, and the cones of effective $1$-cycles $NE(A)$ and $NE(M)$ coincide, where $A$ is the image of $\sA$ under the first reduction map. We also show that there exists a new polarization $H$ on $M$ and our main result is proving that the usual adjunction process on $(M,H)$ terminates, this leading to a coarse classification of $(\sM,\sL)$.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
