Non-defectivity of Grassmannians of planes

Let $Gr(k,n)$ be the Plücker embedding of the Grassmann variety of projective $k$-planes in $\mathbb P^n$. For a projective variety $X$, let $\sigma _s(X)$ denote the variety of its secant $(s-1)$-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 $\bigwedge \nolimits ^{3} {\mathbb C}^{n+1}$. Finally, we classify all defective $\sigma _s(Gr(k,n))$ for $s\le 6$ and provide geometric arguments underlying each defective case.