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Some properties of dual varieties and their applications in projective geometry. (English) Zbl 0793.14026

Algebraic geometry, Proc. US-USSR Symp., Chicago/IL (USA) 1989, Lect. Notes Math. 1479, 273-280 (1991).
[For the entire collection see Zbl 0742.00065.]
The author surveys recent results on dual varieties and offers a list of interesting related open problems. Let \(X \subseteq \mathbb{P}^ n\) be a complex \(n\)-fold; denote by \(X^* \subseteq \mathbb{P}^ N\) its dual variety; put \(n^*=\dim X^*\) and \(\text{codeg} X=\deg X^*\). After recalling the known classification results of \(n\)-folds with small \(n^*\) (i.e. with small dual varieties), the author outlines the problem of classifying \(n\)-folds of given (small) codegree. In particular, he gives a sketch of proof of the classification theorem for codegree three [F. L. Zak in Geometry of complex projective varieties, Cetraro 1990, Mediterranean, Rende, 303-320 (1993)].
The second part of the paper is focused on the links between duality and the notion of extendability. An \(n\)-fold \(X \subseteq \mathbb{P}^ N\) is said to be projectively extendable if there exists a (possibly singular) \((n+1)\)-dimensional variety \(X' \subseteq \mathbb{P}^{N+1}\) (not a cone) of which \(X\) is a hyperplane section. Interesting results on nonextendability (and on the more general notion of non-\(r\)- extendability) are established; in particular, the connection with dual varieties relies on the following criterion: If \(X^*\) is normal, then \(X\) is nonextendable.

MSC:

14J10 Families, moduli, classification: algebraic theory
14N05 Projective techniques in algebraic geometry

Citations:

Zbl 0742.00065
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