How to show that the Grassmannian is irreducible as a variety?
I have no idea.
2026-03-28 10:24:19.1774693459
How to show that the Grassmannian is irreducible as a variety?
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Here's one way: Note that there is a transitive group action on the Grassmannian. The group $GL(n)$ act transitively on the Grassmannian. So choose a smooth point $P \in \mathbb{G}r(k,n)$. Then by acting on $P$ with $GL(n)$, we conclude that all points are smooth and that the Grassmannian is connected.
But smooth connected varieties are irreducible.