On the dimension of secant varieties
Consider an irreducible, projective non-degenerate variety \(X \subset \mathbb{P}^N\). Basic projective invariants related with \(X \subset \mathbb{P}^N\) are its \(k\)-secant varieties which are the Zariski closure of the linear spaces of dimension \(k\) in \(\mathbb{P}^N\) spanned by \(k+1\) points on \(X\). The study of the secant varieties of \(X \subset \mathbb{P}^N\) is relevant to understand the geometry, the topology and the equations defining \(X \subset \mathbb{P}^N\). As usual in projective geometry a notion of expected dimension of the \(k\)-secant variety is known and the problem of classification of \textit{defective} varieties (the ones not reaching the expected dimension) is of interest. In the introduction of the paper under review precise references on this topic are presented. A main ingredient is the so-called Zak's theorem on tangencies bounding the dimension of the subvariety along which a linear spaces is tangent to \(X \subset \mathbb{P}^N\). The paper proposes a generalization of this theorem (see Theorem 6.1) by introducing the notion of \(J_k\)-tangency which generalizes the one of \(J\)-tangency (now \(J_1\)-tangency) of Zak. The authors consider limits of linear spaces of dimension \(s \leq k\) instead of just limits of lines (go to Definition 5.1 for the precise definition). Under suitable regularity conditions they can use this notions of \(J_k\)-tangency and the theorem of tangencies to extend Zak's theorem on linear normality (see Theorem 7.5), to revisit the classification of Scorza varieties in these terms and to make some speculations on possible generalizations of Hartshorne's conjecture on complete intersections.
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