STUDY ON THE COMPLEX AFFINE SPACE CィイD1NィエD1 AND ITS COMPACTIFICATION

我们投资projective compactifications or non-projective Moishezon compactifications ofC - D12 - D1 and the classification of minimal normal compactifications of C - D12 - D1/G，where G is a small finite subgroup of the general linear group GL（2，C）我们将有更多的新结果，我们将有更多的追随者，有六种类型的项目compactifications of c13 D13 with second Betti number equal to one. This was obtained byFurushima before. In this research，we gave a concrete construction of these six compactifications of C3 from the well-knowncompactifications (P - D13 - D1,P - D12 - D1), that is，we gave an explicit birational map of P i D13 D1 to X which is biregular on C i D13 D1-part. Thisfinishes the projective classifications of such compactifications of c13 D13 . next，we also studied the structure of the non-projective compactifications (X，Y) of Csecond Betti number equal to one. In this case这是很容易看到的，canonical divisor可以作为作家遵循：KX=-rY (r>0 is an integer). In this research，我们可以show that the integer r is equal to one or two and that there are many new examples of suchnon-projective compactifications of cai D13 D1.Furthermore，we find that some technique developed in the study of compactifications of c13 we find that some technique developed in the study of compactifications of c13 we find that some technique developed in the study of compactifications of c13to the classification of the minimal normal compactifications of C - 1/G，那么我们succeeded in its classification。

We investigated projective compactifications or non-projective Moishezon compactifications of CィイD13ィエD1 and the classification of minimal normal compactifications of CィイD12ィエD1/G, where G is a small finite subgroup of the general linear group GL(2,C), and we obtained several new results. We will state as follows. There are six types of projective compactifications of CィイD13ィエD1 with second Betti number equal to one. This was obtained by Furushima before. In this research, we gave a concrete construction of these six compactifications of C3 from the well-known compactifications (PィイD13ィエD1,PィイD12ィエD1), that is, we gave an explicit birational map of PィイD13ィエD1 to X which is biregular on CィイD13ィエD1-part. This finishes the projective classifications of such compactifications of CィイD13ィエD1.Next, we also studied the structure of the non-projective compactifications (X,Y) of CィイD13ィエD1 with second Betti number equal to one. In this case, it is easy to see that the canonical divisor can be written as follow: KX=-rY (r>0 is an integer). In this research, we can show that the integer r is equal to one or two and that there are many new examples of such non-projective compactifications of CィイD13ィエD1.Furthermore, we find that some technique developed in the study of compactifications of CィイD13ィエD1 can be applied to the classification of the minimal normal compactifications of CィイD12ィエD1/G, then we succeeded in its classification.