Weakly defective varieties. (English) Zbl 1045.14022
Let \(X\) be a projective variety of dimension \(n\) in \(\mathbb{P}^n\). \(X\) is said to be \(k\)-weakly defective if the general \((k+1)\)-tangent hyperplane to \(X\) has a contact variety of positive dimension.
If \(X\) is \(k\)-defective, i.e. the dimension of its \(k\)-secant variety is strictly less than the expected one, \(\min(r, n(k+1)+k)\), then it is also \(k\)-weakly defective. This suggests that a classification of weakly defective varieties should enlight that of defective varieties. The authors give classification results on \(k\)-weakly defective surfaces both in the case when the surface is \(k\)-defective and in the case when the surface is not \(k\)-defective.
If \(X\) is \(k\)-defective, i.e. the dimension of its \(k\)-secant variety is strictly less than the expected one, \(\min(r, n(k+1)+k)\), then it is also \(k\)-weakly defective. This suggests that a classification of weakly defective varieties should enlight that of defective varieties. The authors give classification results on \(k\)-weakly defective surfaces both in the case when the surface is \(k\)-defective and in the case when the surface is not \(k\)-defective.
Reviewer: M. Palleschi (Milano)
References:
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