The rank of $\alpha\in \bigwedge^2 \mathbb{C}^n$ is the minimal number of decomposable 2-vectors such that $\alpha$ can be written as a sum of them, see https://en.wikipedia.org/wiki/Exterior_algebra . Consider the projectivization $\mathbb{P}(\bigwedge^2 \mathbb{C}^n)$: what is the dimension of the locus of classes of vectors of a given rank?
We clearly have the Grassmannian, whose dimension is $2(n-2)$, which is the locus of vectors of rank one. Moreover, the maximal rank is $k$ if $n=2k$ or $2k+1$. I think the locus of vectors of maximal rank should be an open subset of $\bigwedge^2 \mathbb{C}^n$. How about the other cases? What is the link with the $PGL(n)$-orbits in $\bigwedge^2 \mathbb{C}^n$?
Thank you!
Let $V = \mathbb{C}^n$. If $\alpha \in \mathbb{P}(\wedge^2V)$ has rank $k$ (though I would rather say it is $2k$ in this case) then there is a unique vector subspace $U \subset V$ of dimension $2k$ such that $\alpha$ is in the image of $\mathbb{P}(\wedge^2U) \to \mathbb{P}(\wedge^2V)$. This shows that the locus of rank-$k$ bivectors is isomorphic to an open subspace of the projective bundle $$ \mathbb{P}_{\mathrm{Gr}(2k,V)}(\wedge^2\mathcal{U}), $$ where $\mathcal{U}$ is the tautological bundle. Its dimension, therefore, is $$ 2k(n-2k) + k(2k-1) - 1. $$ And indeed, these loci are precisely the orbits of $\mathrm{PGL}(V)$.