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I'm working the following question:

Let $$\Sigma = \{(L, p) \in G(k,n) \times \mathbb{P}^{n-1} \mid L\subset \mathbb{P}^{n-1}, p \in L\}.$$ Here we're viewing $G(k,n)$ as $(k-1)$-dimensional linear subspaces of $\mathbb{P}^{n-1}$. Show that $\Sigma$ is a projective variety.

A (flawed) attempt:

The Grassmannian $G(k,n)$ is itself a (projective) variety and so it has an open cover by affine varieties, say $\{U_i\}_{i\in I}$ where each $U_i$ is an affine open set isomorphic to an affine variety. (The $U_i$s are described in Construction 8.15 here).

I initially wanted to describe a cover of $\Sigma$ by $\{U_i \times \mathbb{P}^{n-1}\}_{i \in I}$. But this really gives a cover of $G(k,n) \times \mathbb{P^n}$, which is larger than what we want.

Question(s):

Can I modify what I have to give an open cover of $\Sigma$ by affines? From there, I just need to show the diagonal is closed to see that $\Sigma$ is a variety.

Alternatively, is a straight-forward way to see how $\Sigma$ is the vanishing set of some homogenous polynomials, and see that $\Sigma$ is a projective variety that way?

user10039910
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1 Answers1

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The map $\pi : \Sigma \to Gr(k,n), (L,p) \mapsto L$ shows that $\Sigma$ is a $\Bbb P^{k-1}$-bundle over $Gr(k,n)$.

Such varieties are projective, see here for example.

Edit : here is a more elementary way. Take coordinates $x_1, \dots, x_n$ on $\Bbb P^{n-1}$. Consider the Plucker embedding $i : Gr(k,n) \to \Bbb P^m$. This is a closed immersion, so the image is a variety described by equations $f_i = 0$. Now, we would like to embedd $\Sigma$ in $\Bbb P^m \times \Bbb P^{n-1}$, with coordinates $(p,y)$. We take the equations $f_i = 0$ (to be sure that $p \in i(Gr(k,n))$ and we need new equations. Recall that the Plucker coordinates of a $k-1$ plane (represented by a $k \times n$ matrix) is just the corresponding minors of the matrix.

So let $x \in Gr(k,n)$, spanned by $x_1, \dots, x_k$. We have $y \in x$ if and only if the matrix with rows $x_1, \dots, x_n, y$ has all $k+1$ minors which vanishes. Expanding this, you get a polynomials on the form $\sum_i y_i p_{l_i}$ for some indices $l_i$ : this gives you the equations we were looking for ! Now we just compose with the Segre embedding and we are done.

Nicolas Hemelsoet
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