On classification of non-Gorenstein $Q$-Fano $3$-folds of Fano index $1$
On classification of non-Gorenstein $Q$-Fano $3$-folds of Fano index $1$
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关于非 Gorenstein $Q$-Fano $3$-Fano 指数 $1$ 折叠的分类
DOI：
10.2969/jmsj/04720369
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发表时间：
1995
影响因子：
0.7
通讯作者：
Takeshi Sano
中科院分区：
文献类型：
--
作者：
Takeshi Sano
定义1.1。如果一个$d$维的正规复射射角$X$只有终端奇点且反正则Weil因子$-K_{X}$是充足的，则称为$Q$-Fano -fold（参见[KMM]）。定义奇点$P$的索引为最小的正整数$i_{P}$，使得$i_{P}K_{X}$是$P$附近的卡地亚除数。奇点指数为1的奇点称为戈伦斯坦奇点。定义$X$的奇异指数$I(X)$为使$IK_{X}$为卡地亚除数的最小正整数。因此有一个正整数$r$和一个卡地亚因子$H$使得$-IK_{X}\sim rH$。取这样的$r$的最大数目，我们称$r/ $ I$为$X$的Fano指数。
DEFINITION 1.1. A $d$ -dimensional normal complex projective variety $X$ is called a $Q$-Fano -fold if it has only terminal singularities and the anti-canonical Weil divisor $-K_{X}$ is ample (cf. [KMM]). The index of singular point $P$ is defined to be the smallest positive integer $i_{p}$ such that $i_{p}K_{X}$ is a Cartier divisor near $p$ . A singular point of singularity index one is called Gorenstein singularity. Singularity index $I(X)$ of $X$ is defined to be the smallest positive integer such that $IK_{X}$ is a Cartier divisor. Hence there is a positive integer $r$ and a Cartier divisor $H$ such that $-IK_{X}\sim rH$. Taking the largest number of such $r$ , we call $r/I$ the Fano index of $X$ .