Non-Defectivity of Grassmannians of planes

TL;DR

This paper investigates the dimensions of secant varieties of Grassmannians of planes, establishing conditions for expected dimension, classifying defective cases, and connecting results to tensor rank asymptotics.

Contribution

It introduces functions determining when secant varieties of Grassmannians have expected dimension and classifies all defective cases for small s.

Findings

Secant varieties of Gr(2,n) have expected dimension under certain bounds.

Asymptotic behavior of typical tensor rank in wedge powers.

Complete classification of defectivity for s ≤ 6.

Abstract

Let $Gr(k,n)$ be the Pl\"ucker embedding of the Grassmann variety of projective $k$-planes in $\P n$. For a projective variety $X$, let $\sigma_s(X)$ denote the variety of its $s-1$ secant planes. More precisely, $\sigma_s(X)$ denotes the Zariski closure of the union of linear spans of $s$-tuples of points lying on $X$. We exhibit two functions $s_0(n)\le s_1(n)$ such that $\sigma_s(Gr(2,n))$ has the expected dimension whenever $n\geq 9$ and either $s\le s_0(n)$ or $s_1(n)\le s$. Both $s_0(n)$ and $s_1(n)$ are asymptotic to $\frac{n^2}{18}$. This yields, asymptotically, the typical rank of an element of $\wedge^{3} 1pt {\mathbb C}^{n+1}$. Finally, we classify all defective $\sigma_s(Gr(k,n))$ for $s\le 6$ and provide geometric arguments underlying each defective case.