Zak's theorem on superadditivity
The authors give a proof of Zak's theorem: Let \(X\) be a nonsingular, closed subvariety (reduced, irreducible) in \(\mathbb{P}^ N\). Assume that the \(p\)-secant variety \(S^ k (X)\) is not equal to \(\mathbb{P}^ N\). Then \[ \delta_{p+1} (X)\geq \delta_ p (X)+ \delta_ 1 (X), \] where the numbers \[ \delta_ k (X)= \dim S^{k-1} (X)+ \dim X+1- \dim S^ k (X) \] are the secant defects of \(X\) in \(\mathbb{P}^ N\). The proof differs from that of Zak in that it is of an infinitesimal nature. It grew out of an attempt to understand and clarify the proof given by Zak. An earlier such attempt was made by B. Fantechi who gave a proof of the same theorem following closer to the one indicated by Zak. - - The article also contains some results on a more general statement, often called Zak's conjecture.
scientific article; zbMATH DE number 605119