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Varieties with low dimensional dual variety. (English) Zbl 0908.14019

Summary: We classify the complex projective \(n\)-folds \(X \subset \mathbb{P}^N\) with dimension equal to the dimension of its dual \(X^*\) minus one, \(n\leq 2/3N\) and positive defect. We show that such \(n\)-folds are hyperplane sections of the \(n+1\)-folds with \(\dim X= \dim X^*\) (classified by Ein) and some scrolls over curves. The natural generalization is to consider the set \(S_p\) of positive defect \(n\)-folds with \(\dim (X) =\dim (X^*)- (p-1)\) \((p\) a fixed positive integer). We show that the set of possible values of the pair \((N,n)\) with \(n\leq 2/3N\) corresponding to nondegenerate \(n\)-folds in \({\mathcal S}_p\) which are not scrolls is finite.

MSC:

14M07 Low codimension problems in algebraic geometry
14N05 Projective techniques in algebraic geometry
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References:

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