2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216450Let $Gr(k,n)$ be the Plücker embedding of the Grassmann variety of projective $k$-planes in $¶n$. For a projective variety $X$, let $σ_s(X)$ denote the variety of its $s-1$ secant planes. More precisely, $σ_s(X)$ 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 $σ_s(Gr(2,n))$ 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 $σ_s(Gr(k,n))$ for $s\le 6$ and provide geometric arguments underlying each defective case.17 pagesAlgebraic GeometryCommutative Algebra15A69, 15A72, 14Q99, 14M12, 14M99text