An affirmative answer to a question of Ciliberto (Q1320384)

Here (answering in particular a question posed in print by \textit{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 \(\quad\text{Tan} (G(m,1)) = \text{Sec} (G(m,1)))\).