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

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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.