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

We investigated projective compactifications or non-projective Moishezon compactifications of C-D13-D1 and the classification of C-D12-D1/G， where G is a small finite subgroup of the general linear group GL（2，C）， and we obtained several new results。我们将继续前进。There are six types of projective compactifications of CイD13イエD1 with second Betti number equal to one。这是福鲁西岛以前拍的。在此研究中， we gave a concrete construction of these six compactifications of C3 from the well-known compactifications （Py D13 y D1，Py D12 y D1）， that is， we gave an explicit birational map of Py D13 y D1 to X which is biregular on Cy D13 y D1-part。这将完成C-D13-D1的这一预测性分类，下一步，我们还研究了C-D13-D1的非预测性竞争性结构（X，Y）与第二个贝蒂数等于一个。在这种情况下，很容易看出，规范性分流可以按以下方式编写： KX=-rY（r>0是整数）。在此研究中，我们可以展示that the integer r is equal to one or two，that there are many new examples of such non-projective compactications of C-D13-D1.Furthermore， we find that some technique developed in the study of compactifications of C-D13-D1 can be applied to the classification of the minimal normal compactications of C-D12-D1/G，然后我们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.