Conceptual algebraic proof that Grassmannian is closed in Plücker embedding

Question

I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are closed.

Then, I want to talk about Grassmannians. I want to show that the Grassmannian is closed in the Plücker embedding. In other words, I want to know that the set of rank $1$ tensors is closed in $\mathbb{P} \left( \bigwedge\nolimits^k \mathbb{C}^n \right)$. This sounds like it should be a great application of the theorem that projective maps are closed, right? Send $\mathbb{P}^{n-1} \times \mathbb{P}^{n-1} \times \dotsb \times \mathbb{P}^{n-1}$ to $\mathbb{P} \left( \bigwedge\nolimits^k \mathbb{C}^n \right)$ by $(v_1, v_2,\dotsc, v_k) \mapsto v_1 \wedge v_2 \wedge \dotsb \wedge v_k$? Except that this is only a rationally defined map — it isn't defined when the $v_i$ are linearly dependent.

Of course, I can prove that the rank $1$ tensors are closed by brute force.

I also know some conceptual explanations that are not appropriate as the primary proof for this class (though I may well comment on some of them):

• Working topologically over $\mathbb{C}$, every $k$-plane has an orthonormal basis, so $U(n)$ acts transitively on $G(k,n)$ and $G(k,n)$ is compact.

• In terms of the valuative criterion for closedness, it is enough to take a $k \times n$ matrix with entries valued in a dvr and rank $k$ at the generic point and write down its limit in $G(k,n)$. We can do this explicitly in terms of Smith normal form.

• When a reductive group $G$ acts on an affine variety $X$, the map from the semistable points to Proj of the invariant ring is surjective. Apply this with $G = \operatorname{GL}_k$ and $X$ the $k \times n$ matrices. I suspect I am also implicitly using that the ring of invariants is generated by the $k \times k$ minors, which is already nontrivial.

Is there some clever algebraic proof I'm missing, ideally one which uses that projective morphisms are closed?

For context, this is a mixed grad–undergrad course, taught out of Shavarevich volume 1. Everything is over an algebraically closed field and done in a fairly concrete way.

Answer 1

Here is the argument I have written up in my thesis. It was suggested to me by my advisor Jarod Alper. We use the fact that a proper monomorphism is a closed immersion (EGA IV, 18.12.6). Furthermore, we will also use the functor of points perspective of the Grassmannian. Now because the Grassmannian is proper and projective space is separated, the Plücker embedding is proper (this is the property $\mathscr{P}$ exercise in Ravi Vakil's notes). Thus we just need to show it's a monomorphism. However, because :

1. The Grassmannian and projective space are covered respectively by the open subfunctors $\mathbf{Gr}(d,n)_I$ and $\Bbb{P}(\bigwedge\nolimits^{\!d} \mathcal{O}_{\operatorname{Spec} \Bbb{Z}}^{\oplus n})_I$.
2. An endomorphism of a vector bundle $\varphi : \mathcal{E} \to \mathcal{E}$ is an isomorphism iff $\det \varphi$ is;

it is enough to show the base change $$ \mathbf{Gr}(d,n)_I \to \Bbb{P}\big(\bigwedge\nolimits^{\!d} \mathcal{O}_{\operatorname{Spec} \Bbb{Z}}^{\oplus n}\big)_I$$ is a monomorphism. One can view fact (2) above as the $\mathcal{O}_X$-module analog of the fact that a matrix is invertible iff its determinant is non-zero.

Now given a scheme $X$ and quotient $\mathcal{O}_{X}^{\oplus n} \twoheadrightarrow \mathcal{F}$, we may always replace it with a quotient $\mathcal{O}_X^{\oplus n} \twoheadrightarrow \mathcal{O}_X^I$ such that the composition $\mathcal{O}_X^I \hookrightarrow \mathcal{O}_X^{\oplus n} \twoheadrightarrow \mathcal{O}_X^I$ is the identity. This follows from the definition of when two quotients are equal in $\mathbf{Gr}(d,n)_I(X)$. Henceforth, we will only work with quotients $M : \mathcal{O}_X^{\oplus n} \twoheadrightarrow \mathcal{O}_X^I$ of the following form: $M$ is a matrix (with coefficients in $\Gamma(X,\mathcal{O}_X)$) such that the $d$ columns of $M$ corresponding to the subset $I$ are equal to the columns of the $d \times d$ identity matrix. Now let $X$ be a scheme and suppose there are two quotients $M,M' \in \mathbf{Gr}(d,n)_I(X)$ such that $\bigwedge^d M$ and $\bigwedge^d M'$ are equal. This by definition means we have a diagram

$$\require{AMScd}\begin{CD} \bigwedge\nolimits^d\mathcal O_X^I @>>> \bigwedge\nolimits^d\mathcal O_X^{\oplus n} @>\bigwedge\nolimits^d M>> \bigwedge\nolimits^d\mathcal O_X^I \\ @| @| @AA\cong A \\ \bigwedge\nolimits^d\mathcal O_X^I @>>> \bigwedge\nolimits^d\mathcal O_X^{\oplus n} @>\bigwedge\nolimits^d M'>> \bigwedge\nolimits^d\mathcal O_X^{I'} \\ \end{CD}$$

where the top, bottom rows are the identity and so is the left most column. Thus the displayed isomorphism is an honest equality. But now this means that the $d \times d$ minors of $M$ and $M'$ are equal and hence $M = M'$. In other words, the map $\mathbf{Gr}(d,n)_I \to \Bbb{P}(\bigwedge^d \mathcal{O}_{\operatorname{Spec} \Bbb{Z}}^{\oplus n})_I$ is a monomorphism as claimed.

Answer 2

Here are some details for what I think is a variant of Lev Borisov's proposal involving the Plücker relations in the comments. I think if your goal is just to show that the Grassmannian is closed then this is strictly more work than the argument darij links to involving the rank of $\varphi(\omega)$. But in any case the argument avoids the difficulties you mentioned in the comments.

Let $V$ be a finite-dimensional vector space of dimension $n$ and let $\omega \in \bigwedge^k V$. Wedging with $\omega$ gives a map

$$\varphi(\omega) : V \ni v \mapsto \omega \wedge v \in \bigwedge^{k+1} V.$$

Now consider the nondegenerate pairing $\wedge : \bigwedge^k V \otimes \bigwedge^{n-k} V \to \bigwedge^n V$ and pick a nonzero element of $\bigwedge^n V$, or equivalently an identification $\bigwedge^n V \cong 1$ ($1$ the underlying field). This gives us an identification $\bigwedge^k V \cong (\bigwedge^{n-k} V)^{\ast}$. Let $\omega^{\ast} \in (\bigwedge^{n-k} V)^{\ast} \cong \bigwedge^{n-k} V^{\ast}$ be the image of $\omega$ under this identification. Wedging with $\omega^{\ast}$ gives a map

$$\varphi(\omega^{\ast}) : V^{\ast} \ni f \mapsto \omega^{\ast} \wedge f \in \bigwedge^{n-k+1} V^{\ast}.$$

Dualizing this map gives a map

$$\varphi(\omega^{\ast})^{\ast} : \bigwedge^{n-k+1} V \to V$$

which we can now compose with $\varphi(\omega)$, getting a map

$$\omega \wedge \varphi(\omega^{\ast})^{\ast} : \bigwedge^{n-k+1} V \to \bigwedge^{k+1} V.$$

Claim: $\omega$ is a pure tensor iff this map is $0$.

(Expanding this out in a basis gives the Plücker relations, at least if I'm reading these notes correctly.)

Proof. $\Rightarrow$: let $v_1, v_2, \dots v_n$ be a basis with respect to which $\omega = v_1 \wedge \dots \wedge v_k$. Identify $\bigwedge^n V$ with $1$ using $v_1 \wedge \dots \wedge v_n$. Let $v_1^{\ast}, v_2^{\ast}, \dots v_n^{\ast}$ be the dual basis. Then $\omega^{\ast} = v_{k+1}^{\ast} \wedge \dots \wedge v_n^{\ast}$, so the image of $\varphi(\omega^{\ast})^{\ast}$ lies in $\text{span}(v_1, v_2, \dots v_k)$, which wedging with $\omega$ annihilates.

$\Leftarrow$: if $\omega \neq 0$, then $\omega^{\ast} \neq 0$, hence $\varphi(\omega^{\ast}) \neq 0$, hence $\varphi(\omega^{\ast})^{\ast} \neq 0$. Let $v_1, v_2, \dots v_i$ be a basis of $\text{im}(\varphi(\omega^{\ast})^{\ast})$ and complete it to a basis $v_1, v_2, \dots v_n$ of $V$. Again, let $v_1^{\ast}, v_2^{\ast}, \dots v_n^{\ast}$ denote the dual basis.

By hypothesis, $\omega \wedge v_1, \omega \wedge v_2, \dots \omega \wedge v_i = 0$, hence there is some $\omega' \in \bigwedge^{k-i} V$ such that

$$\omega = \omega' \wedge v_1 \wedge \dots \wedge v_i.$$

If $i \ge k$ (in which case this argument shows that $i = k$) then we are done. Otherwise, there must be some $j \in \{ i+1, i+2, \dots n \}$ such that $\omega^{\ast}$ is not divisible by $v_j^{\ast}$, hence such that $\varphi(\omega^{\ast}) v_j^{\ast} \neq 0$. But then there must exist some $\alpha \in \bigwedge^{n-k+1} V$ such that $\alpha(\varphi(\omega^{\ast}) v_j^{\ast}) \neq 0$, hence there must exist some element of $\text{im}(\varphi(\omega^{\ast})^{\ast})$ (namely $\varphi(\omega^{\ast})^{\ast} \alpha$) on which $v_j^{\ast}$ evaluates to something nonzero. But this contradicts $\text{im}(\varphi(\omega^{\ast})^{\ast}) = \text{span}(v_1, v_2, \dots v_i)$.

Hence $i = k$ as desired. $\Box$

Answer 3

One could argue as follows: if $G$ is a connected algebraic group acting (algebraically) on a vector space $W$, then orbits of $G$ of minimal dimension in ${\mathbb P}(W)$ are closed (by the dominant mapping theorem). Now take $G=GL(V)$ and consider an orbit $O$ of minimal dimension in ${\mathbb P}(\wedge ^ k V)$. The diagonals $T$ in $GL(V)$ have fixed point $p$ in $O$ (an easy version of the Borel fixed point theorem). That is $p$ is an eigenvector for $T$ in $\wedge ^k V$; but the only eigenvectors of $T$ are of the form $p_0= v_1\wedge \cdots \wedge v_k$ where $v_i$ are in the stadrard basis. Hence the orbit through $p_0$ is closed. This orbit is precisely the Grassmannian.

Answer 4

Instead of taking $(P^{n-1})^k \dashrightarrow P(\Lambda^kC^n)$ you can argue inductively (on $k$) by consider the map $$ F(k-1,k) = P_{Gr(k-1,n)}(O^n/U_{k-1}) \to Gr(k,n), $$ where $U_{k-1}$ is the tautological bundle on $Gr(k-1,n)$.

Answer 5

Here is a fairly explicit proof using that projective maps are closed, you may enjoy it if you are following Shavarevich book. It uses the following techniques:

• Projective maps are closed
• Each curve has a normalisation, which is also treated by Shavarevich
• Algebro-geometric computation of limits, that is, working in the local ring of a point
• Algorithm to compute the Hermite normal form of a matrix with coefficients in a euclidean ring

The last algorithm is an adaptation of Gauss elimination algorithm for euclidean rings, it is very cleanly described in Jacobson's Algebra I, see finitely generated modules over a pid.

Let $X = \mathbb{P}^{n-1} \times \mathbb{P}^{n-1} \times \cdots \times \mathbb{P}^{n-1}$ and $X^o$ be the open subset of $X$ consisting of $k$-uples of independant lines and let $Y = \mathbb{P} \left( \bigwedge\nolimits^k \mathbb{C}^n \right)$. I denote by $\phi : X^o \to Y$ the naive map from $X^o$ to $Y$ you described in your question.

First we construct the rational image of the map

The image $Z$ of $\phi$ is its image as a rational map, we therefore follow the standard construction: let $\Gamma^o\subset X\times Y$ be the graph of $\phi$ and $\Gamma$ its closure in $X\times Y$. We let $Z$ be the projection of $\Gamma$ in $Z$, which is closed because of the theorem you would like to illustrate.

Then we show that the rational image coincides with the regular image

Let $Z^o = \phi(X^o)$ the image of the map and let us how that $Z^o = Z$ i.e that the Grassmann variety is closed. For any point in $Z$ there is curve $C^o \subset \Gamma^o$ whose closure $C$ meets $X\times\{[z]\}$ — that, is, such that the projection of $C$ contains $[z]$. Let $([v_1], \ldots, [v_k], [z])$ be such a point of $C$.

We now do an infinitesimal analysis around this point to show that $[z]$ belongs to $Z^o$. The key tool of our analysis is the Gauss-Hermite algorithm, which we apply to some matrix with coefficients in local ring of a thickening of $[z]$ in $C$ — or a normalisation of $C$ since we need this ring to be euclidean.

We can find a hyperplane in $\mathbb{P}^{n-1}$ avoiding the various $[v_i]$s, this yields a cell $E = \mathbb{C}^{n-1}$, so that thickening the point $([v_1], \ldots, [v_k], [z])$ yields $k$ vectors $v_i(\epsilon)$ in $E\otimes \mathbb{C}[\epsilon]$ where $\epsilon$ is the local parameter of our thickening.

Applying Gauss-Hermite algorithm yields vectors $\epsilon^{a_i}(w_i +o(\epsilon))$ such that $$ v_1(\epsilon) \wedge \cdots \wedge v_j(\epsilon) = \epsilon^{a_1}(w_1 +o(\epsilon))\wedge\cdots\wedge \epsilon^{a_j}(w_j +o(\epsilon)) $$ for all $j$. We conclude that $[z] = [w_1\wedge\cdots\wedge w_k]$, that is, $Z^o = Z$.

Answer 6

Here is one more approach, I think it is different from the others.

Inside $\mathbb{P} \left( \bigwedge\nolimits^k V \right) \times \mathbb{P} V$, let $X$ denote $\{ (\omega, v) : \omega \wedge v = 0 \}$. This is clearly closed. Consider the projection $X \to \mathbb{P} \left( \bigwedge\nolimits^k V \right)$. Fiber dimension is upper semicontinuous, so let $d-1$ be the greatest fiber dimension and let $\Gamma$ be the closed locus in $\mathbb{P} \left( \bigwedge\nolimits^k V \right)$ where that greatest dimension is achieved.

Looking at a rank one point of $\mathbb{P} \left( \bigwedge\nolimits^k V \right)$, $d \geq k$. For $x \in \Gamma$, let $V_x \subset V$ be the subspace for which $\mathbb{P}(V)$ is the fiber. So $d = \dim V_x$, choose a basis $v_1$, $v_2$, ..., $v_d$. Then $v_1 \wedge \cdots \wedge v_d$ divides $x$, as in several other proofs, so $d \leq k$. This shows that $d=k$ and that every point of $\Gamma$ is of rank $1$, conversely, it is obvious that the fiber above every rank one point is of dimension $k-1$.

Basically, this is Qiaochu's answer/Lev Borsiov's comment, except using semicontinuity of fiber dimension instead of explicit equations.

Answer 7

In the paper

• Mike Eastwood, Peter W. Michor: Some remarks on the Plücker relations. Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 63 (2000), 85-88. pdf of a longer version

there are 5 versions of the Pluecker relations in theorem 1. Versions (3) and (4) might be helpful. All proofs are included.

Answer 8

$\newcommand\bwedge{\bigwedge\nolimits}$This is an old question, but I think the following idea is genuinely different from the other answers and might be interesting to anyone who comes across it. I got this from the second edition of EGA I (Springer, 1971), section 9.8.

The argument is a little convoluted to set up, but the idea is simple (I'll provide details below). Restricting to open subfunctors, we present the morphism $G_m(E)_H \to G_1(\bwedge^m E)_H$ as the graph of another map, and since the codomain is separated it is a closed immersion. To get there, first set $E'=\mathcal{O}_S^H$ and $E \simeq E'\oplus E''$, so that $G_m(E)_H$ is the collection $\operatorname{Hom}_{\mathcal{O}_S}(E'',E')$ of projections onto the factor $E'$. Then decompose $$\bwedge^m (E'\oplus E'') \simeq \bwedge^m E' \oplus \left(\bwedge^{m-1}E'\otimes E''\right) \oplus (\text{junk}), $$ so that $G_1(\bwedge^m E)_H$ comes from projectors of the form $$ \operatorname{Hom}_{\mathcal{O}_S}(\bwedge^{m-1}E'\otimes E'', \bwedge^m E') \times_S (\text{junk}). $$ Then it's straightforward to show the first factor is $G_m(E)$.

I like this argument because it is quite elementary and works in great generality. In the context of this question, it's also useful as it doesn't, at least as a heuristic, rely on abstract varieties. The disadvantage is that it doesn't exhibit the Plücker relations directly, but I don't believe it would be too hard to pull them out.

--- In more detail ---

I will consider the functor $G_m(E)$ of rank $m$ locally free quotients of a given quasicoherent sheaf of $\mathcal{O}_S$-modules $E$. For simplicity we let $E=\mathcal{O}_S^I$ for some index set $I$, the general case reduces to this using functoriality along epimorphisms $E'\to E$.

The Plücker map is associated to the natural transformation $G_m(E) \to G_1(\bwedge^m E)$ whose component at an $S$-scheme $T$ is $$ \{ q:E_{(T)}\to F\} \longmapsto \{\bwedge^m q:\bwedge^m E_{(T)} \to \bwedge^m F\}.$$

As in the other answers we will check this is a closed immersion on the open subfunctor corresponding to a size $m$ subset $H\subset I$. Let $U_H$ represent the subfunctor of those quotients in $G_m(E)$ such that $$ E' = \mathcal{O}_T^H \hookrightarrow E_{(T)} \to F $$ is an isomorphism, and $V_H$ those in $G_1(\bwedge^m E)$ such that $$ \bwedge^m E' \hookrightarrow \bwedge^m E_{(T)} \to H $$ is iso.

We can split $E \simeq E' \oplus E''$, so that $U_H$ is the vector bundle associated to the $\mathcal{O}_S$-module $$ \operatorname{Hom}_{\mathcal{O}_S}(E'',E'). \tag{$*$}\label{480804_star}$$ Then we notice that, since $$\bwedge^m (E'\oplus E'') \simeq \bwedge^m E' \oplus \left(\bwedge^{m-1}E'\otimes E''\right) \oplus (\text{junk}), $$ the subscheme $V_H$ is also a vector bundle on $S$, associated to $$ \operatorname{Hom}_{\mathcal{O}_S}(\bwedge^{m-1}E'\otimes E'', \bwedge^m E') \times_S (\text{junk}). $$ Using the tensor–hom adjunction and the fact that $\operatorname{Hom}(\bwedge^{m-1} E',\bwedge^{m} E') \simeq E'$, we see that the first factor in the previous equation is isomorphic to the module in \eqref{480804_star}. It's also straightforward to check that our natural transformation is the identity on that factor.

All this is to say that the Plücker map $\omega$ restricts to a map $U_H \to V_H \simeq U_H \times V_H'$ (since the Yoneda embedding preserves products), and is the identity on the $U_H$ factor. Since $V_H$ is separated (affine) over $S$, this implies it is a closed immersion.