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An affirmative answer to a question of Ciliberto. (English) Zbl 0805.14026
Here (answering in particular a question posed in print by C. Ciliberto) the author gives a very interesting example of nondegenerate smooth projective submanifold $$X \subset \mathbb{P}^{2n + 1}$$, $$n \geq 9$$, $$\dim (X) = n$$, with $$K_ X$$ ample (hence $$X$$ not ruled by lines), such that the tangent variety $$\text{Tan} (X)$$ has dimension $$\leq 2n-1$$ and such that for a general $$P \in X$$, the embedded tangent space $$T_ PX$$ intersects $$X$$ at some points $$\neq P$$. The key for the construction is the fact that for $$m\geq 3$$ the secant variety $$\text{Sec} (G(m,1))$$ of the Plücker embedding of the Grassmannian $$G(m,1)$$ of lines in $$\mathbb{P}^ m$$ has dimension $$\leq 2 \dim (G(m,1))-3$$ (hence $$\quad\text{Tan} (G(m,1)) = \text{Sec} (G(m,1)))$$.
Reviewer: E.Ballico (Povo)
##### MSC:
 14N05 Projective techniques in algebraic geometry 14M15 Grassmannians, Schubert varieties, flag manifolds
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##### References:
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