3-Dimensional Fano Varieties with Canonical Singularities
3-Dimensional Fano Varieties with Canonical Singularities
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具有正则奇点的 3 维 Fano 簇
DOI：
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发表时间：
1989
期刊：
影响因子：
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通讯作者：
K. Shin
中科院分区：
文献类型：
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作者：
K. Shin
在这篇文章中，所谓簇是指复数域上的不可约射影簇。设X是一个具有典型奇点的三维Fano簇，H是一个Cartier因子，对X的指数r（X）满足-KX simr（X）H.本文的目的是研究有理映射Phi {|H|}$和$的一般成员的奇点|H| $ .特别是在r（X）=2的情况下，这是最基本的，我们发现$Phi_{|H|}$如下。(1)当$d=H^{3}geqq 3$时，闭浸入$P^{d+1}$。(2)当$d=2$时，$P^{3}$上的一个双重覆盖.（3）当d=1时，定义了一个有理映射，该映射的唯一点的闭包是光滑椭圆曲线。此外，一般成员$S$的$|H| $在$Scap Sing（X）$处有有理双点。
In this article by a variety we mean an irreducible reduced projective variety over the field of complex numbers. Let $X$ be a 3-dimensional Fano variety with canonical singularities and $H$ be a Cartier divisor satisfying $-K_{X}sim r(X)H$ for the index $r(X)$ of $X$. The purpose of this article is to study the rational map $Phi_{|H|}$ and singularities of a general member of $|H|$ . In particular in the case of $r(X)=2$ , which is the most essential, we find $Phi_{|H|}$ to be as follows. (1) When $d=H^{3}geqq 3$ , a closed immersion into $P^{d+1}$ . (2) When $d=2$ , a double covering over $P^{3}$ . And (3) when $d=1$ , a rational map that is defined except exactly one point and the closure of whose general fiber is a smooth elliptic curve. And furthermore a general member $S$ of $|H|$ has rational double points at $Scap Sing(X)$ .