A new example of a compactification of \(\mathbb{C}^ 3\)
Let \((X,Y)\) be a smooth projective compactification of \(\mathbb{C}^ 3\) with the second Betti number \(b_ 2(X)=1\). Then \(Y\) is an irreducible ample divisor on \(X\) and the canonical divisor can be written as \(K_ X=-rY\) \((r\in\mathbb{Z},r>0)\), in particular, \(X\) is a Fano 3-fold of the first kind. We call the integer \(r\) the ``index'' of \(X\). In the case of \(r\geq 2\), the structure of \((X,Y)\) is completely determined by the former works of the author, N. Nakayama, T. Peternell and M. Schneider. In the case of \(r=1\), it is known that \(X\) is isomorphic to the Fano 3-fold \(V_{22}\hookrightarrow\mathbb{P}^{13}\) of genus \(g=12\), degree 22 in \(\mathbb{P}^{13}\) and \(Y\) is a nonnormal hyperplaine section of \(V_{22}\). Among the Fano 3-fold of \(g=12\), degree 22 in \(\mathbb{P}^{13}\), there is a special one, which was constructed by Mukai-Umemura. In this paper, we construct a new compactification of \(\mathbb{C}^ 3\) from the example of Mukai-Umemura.
scientific article; zbMATH DE number 549763
scientific article