A remark on Zak's theorem on tangencies (Q2391556)

Let \(X\subset \mathbb {P}^N\) be a smooth and non-degenerate \(n\)-dimensional variety. Let \(S(X) \subseteq \mathbb {P}^N\) be the secant variety of \(X\) and let \(X^\ast \subset \mathbb {P}^{N\ast}\) be the dual variety of \(X\). Set \(n^\ast := \dim X^\ast \), \(s:= \dim S(X)\) and \(c:= N-s\). First the author using the connectedness theorem of [\textit{W. Fulton} and \textit{J. Hansen}, Ann. Math. (2) 110, 159--166 (1979; Zbl 0389.14002)] gives the following reformulation of Zak's theorem of tangency (formally stronger than the original one when \(S(X) \neq \mathbb {P}^N\)) [\textit{F. L. Zak}, Tangents and secants of algebraic varieties. Translations of mathematical monographs, Vol. 127. Providence, RH: American Mathematical Society (1993; Zbl 0795.14018)] he proves the following result: \quad Let \(L\subset \mathbb {P}^N\) be a linear subspace of dimension \(m\) tangent to \(X\) along a closed subvariety of dimension \(r\). Then \( r \leq \min \{m-n,s-1-n\}\). The author shows that his reformulation has very strong consequences for the classification theory of almost extremal varieties. Using also [\textit{L. Ein}, Duke Math. J. 52, 895--907 (1985; Zbl 0603.14026)] and [\textit{P. Ionescu} and \textit{F. Russo}, Am. J. Math. 135, No. 2, 349--360 (2013; Zbl 1271.14055)] he proves the following results: \quad (1) The twisted normal bundle is \((s-1-n)\)-ample. \quad (2) \(n^\ast \geq n+c\); if \(c>0\), then \(X^\ast\) is singular. \quad (3) If \(s\leq 2n-1\) (resp. \(s\leq 2n-2\)), then every hyperplane section of \(X\) is reduced (resp. normal). \quad (4) If \(c>0\), then \(n^\ast \geq n+c+1\) with equality if and only if \(X\) is either a curve or \(N=5\) and \(X\) is a Veronese surface. \quad (5) Assume \(n\geq 4\) and \(c>0\). We have \(n^\ast =n+c+2\) if and only if either \(X\) is a scroll over a curve or an isomorphic projection of the Segre embedding \(\mathbb {P}^2\times \mathbb {P}^{n-2} \subset \mathbb {P}^{3n-4}\). \quad (6) Assume that \(X\) is a scroll over a surface, \(c>0\), and \(2n+1-s =2\). Then \(X\) is an isomorphic projection of the Segre embedding \(\mathbb {P}^2\times \mathbb {P}^{n-2} \subset \mathbb {P}^{3n-4}\).