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:
- Using the Plücker embedding, where the subspace spanned by any orthonormal $v_1, v_2, ..., v_n$ is given by $v_1 \wedge v_2 \wedge ... \wedge v_n$
- 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
- The Plücker embedding can be identified with the $n$'th compound matrix $C_n(M)$
- 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:
- Is there some sense in which these two varieties are "the same" variety?
- 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?