A study on compactifications of complex affine planes and spaces

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(1) Research 1: The boundary of compactification of C^2 into Hirzebruch surfacesconsists of two irreducible components, one of which can have a singularity byL.Brenton. Then I can construct more general examples including the Brenton’ s one. Ialso prove that one of the irreducible components of boundary divisor is a smoothrationa curve. Further , I suggested the possibility of a birational geometric proof of thetheorem by Abyankar-Moh- Suzuki. (2) Research 2: I studied the detail structure of singular Fano threefolds withGorenstein terminal singularities as a compactification of C^2.,especially , applying thetheory of smoothing of Fano threefolds with Gorenstein terminal singularities, I candetermine the detail structure of the known examples of singular Fanocompactifications of C^3..