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In "Algebraic geometry: a first course", by Harris, Grassmannian is described, under the Plucker embedding, as the locus of totally decomposable vectors in the projectivization of the exterior power $\bigwedge^k V$. Here

Decomposable elements of $\Lambda^k(V)$

a characterization of $m$-decomposable vectors in $\bigwedge^k V$ is given; however, I am struggling to find any reference for the locus of $m$-decomposable vectors in $\mathbb{P}(\bigwedge^k V)$. I fell like it should be a variety, not necessarily smooth, that contains the Grassmannian $G(k,V)$ for every choice of $m$. I am particularly interested in the case of $1$-decomposable vectors in $\mathbb{P}(\bigwedge^3 \mathbb{C}^5)$, but I think there should exist a fancy treatment of the topic. Can you provide me some reference?

Nutella Warrior
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1 Answers1

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If a 2-vector is 1-decomposable, it is also 2-decomposable (more generally, if a $k$-vector is $(k-1)$-decomposable, it is $k$-decomposable). In particular, the locus of 1-decomposable vectors in $\mathbb{P}(\wedge^2\mathbb{C}^5)$ is $G(2,5)$.

EDIT. On the other hand, the action of the group $\mathrm{PGL}(\mathbb{C}^5)$ on $\mathbb{P}(\wedge^2\mathbb{C}^5)$ has only two orbits, the Grassmannian and its complement, and since the locus of 1-decomposable vectors is obviously $\mathrm{PGL}(\mathbb{C}^5)$-invariant, it follows that it is equal to the entire space $\mathbb{P}(\wedge^2\mathbb{C}^5)$.

Sasha
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  • I am sorry, I wrote $\bigwedge^2 \mathbb{C}^5$ but I am really interested in $\bigwedge^3 \mathbb{C}^5$! Edited. Anyway, I am using the notation in the first answer of the question I linked, so that $G(3,\mathbb{C}^5)$ is the locus of $5$-decomposable vectors, and I am looking for the locus of vectors in the form $v\wedge \eta$ with $v\in \mathbb{C}^5$ and $\eta\in \bigwedge^2\mathbb{C}^5$. Is it clearer now? – Nutella Warrior Feb 16 '20 at 20:57
  • @NutellaWarrior: I added a discussion of this case to my answer. – Sasha Feb 17 '20 at 04:29
  • I see. So in general the locus of $m$-general vector in $\mathbb{P}(\bigwedge^k \mathbb{C}^n)$ correspond to the orbit under the action of PGL(\mathbb{C}^n) of a vector in the form $v_1\wedge \ldots\wedge v_m\wedge \eta$, for $v_i\in \mathbb{C}^n$ and $\eta\in \bigwedge^k \mathbb{C}^{n-m}$ – Nutella Warrior Feb 17 '20 at 06:34
  • @NutellaWarrior: A union of orbits, to be more precise. And $\eta \in \wedge^{k-m}\mathbb{C}^n$. – Sasha Feb 17 '20 at 08:10