Non-Defectivity of Grassmannians of planes

Abstract

Let $Gr(k,n)$ be the Plücker 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 $n\ge 9$ 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.

17 pages

References in corpus (1)

• Secant Varieties of (P ^1) X .... X (P ^1) (n-times) are NOT Defective for n \geq 5