A study on the singular Fano compactifications of 3-dimensional complex affine spaces

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We study the classification of the singular Fano 3-fold with the Picard number one which is a compactification of complex affine space C^3. Let r(0<r<5) be the index of such a Fano compactification. In our study, we can determine the structure for the cases of r=2,3 and 4. Moreover we construct a non-trivial example of the singular Fano compactification of C^3, which will be helpful for the determination of the case of the index r=1. We also get two results in the relation with the compactiication of affine spaces.