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

几年来对弱Fano三重结构的研究告诉我们如下分类(我正在准备一篇题为“弱Fano三重结构与del Pezzo纤丝”的论文)：设V是光滑的弱Fano三重结构，其反正则模型只有终端奇点。假设V有一条D类型的极值射线(具有Del Pezzo纤维的类型)的次数为d。用N(D)表示d次的变形类别的数目。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，并且可以确定属于每个变形类的一般弱Fano三重的结构。D=2时的弱Fano三重可视为加权射影空间丛中的一个因子。这种观点导致了与前一种观点相同的分类。本文研究了加权射影空间丛中的一个因子--光滑弱Fano四重，作为上述三重的情形，本文研究了光滑的弱Fano四重。我们得到了几个这种类型的弱Fano四重的例子。虽然从分类理论的观点来看，这一结果还不够充分，但对于射影空间丛中四重为除数的情形，我们得到了一个分类。参见论文《投影线上的射影空间丛中的弱Fano四重折叠》，经济学和信息研究，2002年。

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.