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Extensions of projective varieties and deformations. I. (English) Zbl 0770.14005

Given a projective variety \(V\subset\mathbb{P}^ n\), a projective variety \(W\subset\mathbb{P}^{n+1}\) is an extension of \(V\) if \(V\) is a hyperplane section of \(W\) in \(\mathbb{P}^{n+1}\). If \(W\) is not a cone over \(V\), \(W\) is a nontrivial extension of \(V\). Write \(\mathbb{P}^ n=\mathbb{P}(E)\) with an \((n+1)\)-dimensional vector space \(E\), where \(\mathbb{P}(E)\) is the set of lines in \(E\). Define a locally free sheaf \(\Gamma_ V\) on \(V\) as \(\Gamma_ V=(P^ 1({\mathcal O}_ V(1))^*\) with \(P^ 1\) denoting the sheaf of principal parts of the first order, or by an exact sequence \[ 0\to\Gamma_ V\to E\otimes{\mathcal O}_ V\to N_{\mathbb{P}(E)| V}(- 1)\to 0, \] where \(N_{\mathbb{P}(E)| V}\) is the normal bundle of \(V\) in \(\mathbb{P}(E)\). Define an integer \(\alpha(B)=\dim\text{Coker}(E\to H^ 0(V,N_{\mathbb{P}(E)| V}(-1))\).
A main theorem of this article then states that if \(V\neq\mathbb{P}^ n\), \(V\) is not a quadric, and \(\alpha(V)=0\) then \(V\) is not a hyperplane section of a projective variety other than a cone. A proof of this result was first obtained by F. L. Zak in 1984 (unpublished work). The present proof depends on deformation theory. A similar result was obtained also by T. Fujita [J. Math. Soc. Japan 34, 355-363 (1982; Zbl 0478.14002].

MSC:

14E25 Embeddings in algebraic geometry
14N05 Projective techniques in algebraic geometry

Citations:

Zbl 0478.14002
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