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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)

MSC:
 14N05 Projective techniques in algebraic geometry 14J40 $$n$$-folds ($$n>4$$)
Keywords:
Severi variety; secant variety
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