A remark on Zak's theorem on tangencies

TL;DR

This paper offers a new formulation of Zak's theorem on tangencies, improving bounds on dual variety dimensions and classifying special cases where secant varieties are not full-dimensional.

Contribution

It introduces a modified version of Zak's theorem and applies it to derive tighter bounds and classifications for dual and secant varieties.

Findings

Improved bound on the dimension of the dual variety

Classification of extremal and next-to-extremal cases

Applications to secant variety dimension analysis

Abstract

We present a slightly different formulation of Zak's theorem on tangencies as well as some applications. In particular, we obtain a better bound on the dimension of the dual variety of a manifold and we classify extremal and next-to-extremal cases when its secant variety does not fill up the ambient projective space.