Tangential projections and secant defective varieties

TL;DR

This paper characterizes certain Veronese embeddings and their projections as unique smooth varieties with specific secant defect properties, extending Zak's classification of Scorza varieties.

Contribution

It provides a new characterization of Veronese embeddings and their projections within Zak's framework, identifying unique geometric properties.

Findings

Characterization of Veronese embeddings of projective space by secant defect

Identification of smooth projections from a point as unique cases

Extension of Zak's classification of Scorza varieties

Abstract

Going one step further in Zak's classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of $\P^n$ given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and non-degenerate projective subvarieties of $\P^N$ that can be projected isomorphically into $\P^{2n}$ when $N\geq\binom{n+2}{2}-2$.