CLASSIFICATION OF WEAK FANO MANIFOLDS WHICH ARE DIVISORS IN THE WEIGHTED PROJECTIVE SPACE BUNDLES OVER THE PROJECTIVE LINE

射影线上加权射影空间束中的弱 FANO 流形的分类

• 批准号: 12640048
• 负责人: TAKEUCHI Kiyohiko
• 金额: $ 0.38万
• 依托单位: Gifu Shotoku Gakuen University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640048/
• 关键词: Fano variety; algebraic variety; the classification theory; anti-canonical model; projective space bundle

The research for several years about weak Fano threefolds tells us the following classification (I am in preparation for the paper titled "Weak Fano threefolds with del Pezzo fibrations"):Let V be a smooth weak Fano threefold whose anti-canonical model has only terminal singularities. Assume that V has an extremal ray of type D (the type having del Pezzo fibration) of degree d. Denote the number of deformation classes for degree d by N(d). Then, we have N(1)=2, N(2)=4, N(3)=7, N(4)=11, N(5)=11, N(8)=9, and N(9)=3, and can determine the structure of the general weak Fano threefold belonging to each deformation class.The problem for the case d=6 is open. The weak Fano threefold for the case d=2 can be regarded as a divisor in weighted projective space bundles. This point of view leads to the same classification as the previous one. See the paper "Weak Fano threefolds with del Pezzo fibration of degree two," Economics and Information Studies (working paper series in our faculty), 2001.This research studied the smooth weak Fano fourfold which is a divisors in weighted projective space bundles as the case of threefolds above. We obtained several examples in weak Fano fourfolds of this type. Although the result is not yet sufficient from the viewpoint of classification theory, a classification was obtained for the case that the fourfold is a divisor in projective space bundles. See the paper "Weak Fano fourfolds in the projective space bundle over the projective line", Economics and Information Studies, 2002.

几年来关于弱Fano三重性的研究告诉我们以下的分类（我正在准备论文“Weak Fano threefolds with del Pezzo fibrations”）：设V是一个光滑的弱Fano三重性，其反正则模型只有终端奇点。假设V有一条D型极值射线（具有del Pezzo纤维化的类型），次数为d。用N（d）表示次数为d的变形类的数量。然后，我们有N（1）=2，N（2）=4，N（3）=7，N（4）=11，N（5）=11，N（8）=9，N（9）=3，可以确定属于每个变形类的一般弱Fano三重结构，d=6的情况是公开问题。在d=2的情形下，弱Fano三重性可以看作是加权射影空间丛中的一个因子.这个观点导致了与前一个相同的分类。参见论文“Weak Fano threefolds with del Pezzo fibration of degree two，”Economics and Information Studies（Working paper series in our faculty），2001.本研究研究了加权射影空间丛中作为因子的光滑弱Fano四重性，作为上述三重性的情形.我们在弱Fano四重中得到了几个例子。虽然从分类论的观点来看，这个结果还不够充分，但对于四重是射影空间丛中的因子的情形，得到了一个分类。参见论文“Weak Fano fourfolds in the projective space bundle over the projective line”，Economics and Information Studies，2002。