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On odd dimensional projective manifolds with smallest secant varieties. (English) Zbl 0887.14029
Let \(X\) be an \(n\)-dimensional nondegenerate projective manifold \(X\) in \(\mathbb{P}^N\) over an algebraically closed field \(k\) of characteristic 0, and \(\text{Sec} X\) the secant variety of \(X\) in \(\mathbb{P}^N\). It is well known that if \(\text{Sec} X \neq\mathbb{P}^N\) then \(\dim \text{Sec} X \geq(3n +2)/2\), that if the quality holds then \(X\) is called a Severi variety, and that Severi varieties are completely classified. In this article, we consider the case when \(\text{Sec} X \neq\mathbb{P}^N\), \(n\) is odd, \(n\geq 3\), and \(\dim \text{Sec} X= (3n+3)/2\), and show that the possible values of dimension of the contact locus of \(\text{Sec} X\) with its general embedded tangent space are \((n+1)/2\) or \((n+5)/2\), and that if the general contact locus has dimension \((n+1)/2\), then \(X\) is a Fano manifold and the possible values of \(n\) are 3, 5, 7, 15 (in these cases there are examples), \(2^m-1 (m\geq 7)\), or \(2^m\cdot 3-1(m\geq 5)\) (in these cases no examples are known). We also determine the manifold \(X\subset\mathbb{P}^N\) when \(n=5\) or \(n=7\) and the general contact locus of \(\text{Sec} X\) has dimension \((n+1)/2\).
Reviewer: M.Ohno (Tokyo)

14N05 Projective techniques in algebraic geometry
14J40 \(n\)-folds (\(n>4\))
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