# Is the Grassmannian a unique algebraic variety?

I'm fairly new to algebraic geometry, so this is may just be fairly simple question about when two varieties are "the same" variety.

In this question, I noted that the Grassmannian can be expressed as an algebraic variety in (at least) two ways:

1. Using the Plücker embedding, where the subspace spanned by any orthonormal $$v_1, v_2, ..., v_n$$ is given by $$v_1 \wedge v_2 \wedge ... \wedge v_n$$
2. Using the projective embedding, where a subspace is identified with the unique orthogonal projection matrix to that subspace

Or as matrices, if the matrix $$M = [v_1 | v_2 | ... | v_n]$$, where the $$v_i$$ are column vectors, then

1. The Plücker embedding can be identified with the $$n$$'th compound matrix $$C_n(M)$$
2. The projective embedding can be identified with the projection matrix $$MM^\dagger$$, where $$M^\dagger$$ is the pseudoinverse

These would appear to be two totally different algebraic varieties, defined by different sets of polynomial equations, embedded in two different Euclidean spaces, with two different metrics. Yet somehow, people often talk of the "Grassmann variety" as though there were a unique variety associated with the Grassmannian. So this leads to my question:

1. Is there some sense in which these two varieties are "the same" variety?
2. In general, what intrinsic properties can one use to determine if two varieties are "the same variety," independent of their particular embedding into a Euclidean space?
• As everywhere in mathematics, there is a notion of isomorphism. A simpler example of your phenomenon is that embedding of $\Bbb P^1$ as a smooth conic in $\Bbb P^2$ or as a twisted cubic in $\Bbb P^3$. Jun 1, 2019 at 0:16
• In what sense are the two varieties isomorphic though? They aren't defined by the same equations nor have the same metric. I guess they'd both be homeomorphic as topological spaces? Is there nothing stronger than that? Jun 1, 2019 at 1:11
• en.wikipedia.org/wiki/Morphism_of_algebraic_varieties Jun 1, 2019 at 3:18