How to think Grassmannian as a projective variety?

Question

I'm just looking for some explanation for the grassmannian as a projective variety and plücker embedding.

Answer 1

This seems prior to your other question so let's start here: the Plücker embedding is a natural construction. I don't want to write an article here so I'll suggest Chapter 11 of Hassett's Introduction to Algebraic Geometry as a reference that develops most of the background from multilinear algebra while constructing the embedding. Another reference, available legally online, is section 6m of Milne's notes. [In case he changes the numbering someday: I am referring to version 6.00]

Briefly — if you do know something about the exterior algebra — the idea is that if $W$ is a $k$-dimensional subspace of an $n$-dimensional vector space $V$ then $\bigwedge^k W$ is naturally a one-dimensional subspace of $\bigwedge^k V$, which is the same as a point in the projective space $\mathbb{P}(\bigwedge^k V)$. The first issue here is checking that this is an injection. The second issue is that not all points of $\mathbb{P}(\bigwedge^k V)$ have this form, so you had better check that the subset of such points is cut out by homogeneous polynomials.

This is actually a little tricky, and finding a generating set for the corresponding ideal is real work. If only to gain some appreciation for this before reading up, it might be worth playing around with the smallest example that isn't a projective space, i.e., the space $G(2, 4) = \mathbb{G}(1,3)$ of $2$-planes in $4$-space. You're trying to characterize the elements of $\bigwedge^2 V$ that can be written as $v_1 \wedge v_2$ with $v_1, v_2 \in V$. In this case a single equation works.