A new construction of a compactification of \({\mathbb{C}}^ 3\)
A smooth projective compactification of \({\mathbb{C}}^ 3\) is a pair (X,Y) with X a smooth projective 3-fold, and Y a subvariety of X such that X-Y is analytically isomorphic to \({\mathbb{C}}^ 3\). In a previous paper [\textit{M. Furushima}, Nagoya Math. J. 104, 1-28 (1986; Zbl 0612.14037)], it is shown that if there is an algebraic isomorphism of X-Y onto \({\mathbb{C}}^ 3\), and Y is normal, then X is a Fano 3-fold of index \(r\) \((r=1,...,4),\) with \(b_ 2(X)=1\), and Y is a hyperplane section of X. Furthermore, a very precise description of the pair (X,Y) can be given for the cases \(r=2, 3, 4\). When \(r=2\), in particular, one can show that \((X,Y)=(V_ 5,H_ 5)\), where \(V_ 5\) is a Fano 3-fold of degree 5 in \({\mathbb{P}}^ 6\), and \(H_ 5\) is a singular Del Pezzo surface with exactly one rational double point x of \(A_ 4\)-type. If one projects \(V_ 5\) from a line C in \(V_ 5\) through x, one can see that \(V_ 5-H_ 5\) is algebraically isomorphic to \({\mathbb{C}}^ 3\). Moreover, if \(H^{\infty}_ 5\) is the ruled surface swept out by lines which intersect C, one can see that \(H^{\infty}_ 5\) is a non-normal hyperplane section of \(V_ 5\) such that \(V_ 5-H^{\infty}_ 5\) is algebraically isomorphic to \({\mathbb{C}}^ 3\). In particular, both these sections of \(V_ 5\) belong to the same linear system \(| H-2x|.\) In the paper under review, the authors study the question of how many members of the linear system \(| H-2x|\) can be normal or non- normal boundaries of \({\mathbb{C}}^ 3\) in \(V_ 5\); the technique they use is the double projection from the singular point x, and, in particular, the authors obtain, in this way, a new construction of a compactification of \({\mathbb{C}}^ 3\), in the case of index r\(=2\).
smooth projective compactification of complex 3-space
scientific article; zbMATH DE number 4152575