A new construction of a compactification of \({\mathbb{C}}^ 3\)
type: Item
instance of: scholarly article
title: A new construction of a compactification of \({\mathbb{C}}^ 3\)
zbMATH Open document ID: 0703.14025
author: M. Furushima
author: Noboru Nakayama
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1989-01-01T00:00:00Z
review text: 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\(=2\).
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 32J05
zbMATH DE Number: 4152575
zbMATH Keywords: smooth projective compactification of complex 3-space
zbMATH Keywords: Fano 3-fold
zbMATH Keywords: singular Del Pezzo surface
reviewed by: D. C. Struppa
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227726
OpenAlex ID: W2167828563
cites work: Singular Del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space C3
cites work: Q4143433
cites work: Anticanonical models of three-dimensional algebraic varieties
cites work: Q3825176
cites work: The birational geometry of surfaces with rational double points
cites work: Vector bundles on complex projective spaces
cites work: Compactifications of \({\mathbb{C}}\) 3. I
cites work: Projective modules over polynomial rings
cites work: Q3222980
DOI: 10.2748/TMJ/1178227726
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description: scientific article; zbMATH DE number 4152575