A note on the tangential envelopes of Veronese varieties (Q1768061)

Let \(X \subseteq \mathbb P^m\) be an integral variety and \(TX \subseteq \mathbb P^m\) its tangent variety. Here the author iterates this process and defines inductively \(T^s(X)\), \(s \geq 2\), as \(T^s(X):= T(T^{s-1}X)\). He computes its dimension for \(s \leq 4\) when \(X\) is the image of \(d\)-Veronese embedding of \(\mathbb P^m\) and shows that quite often this dimension is not the expected one. For arbitrary \(s\) he conjectures the value of this dimension.