A new example of a compactification of \(\mathbb{C}^ 3\)
type: Item
instance of: scholarly article
title: A new example of a compactification of \(\mathbb{C}^ 3\)
zbMATH Open document ID: 0790.32027
DOI: 10.1007/BF02571665
author: M. Furushima
published in: Mathematische Zeitschrift
publication date: 1994-06-20T00:00:00Z
full work available at URL: https://eudml.org/doc/174494
review text: 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.
Mathematics Subject Classification ID: 32J05
Mathematics Subject Classification ID: 32J17
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J30
zbMATH DE Number: 549763
zbMATH Keywords: compactification of \(\mathbb{C}^ 3\)
zbMATH Keywords: Fano 3-fold
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 549763