Associated Varieties and General Resultants

Abstract

The Grassmann variety (or Grassmannian) G(k, n) is the set of all k-dimensional vector subspaces in Cⁿ. For k = 1, this is the projective space P ⁿ⁻¹. Since vector subspaces in Cⁿ correspond to projective subspaces in P ⁿ⁻¹, we see that G(k, n) parametrizes (k−1)-dimensional projective subspaces in P ⁿ⁻¹. In a more invariant fashion, we can start from any finite-dimensional vector space V and construct the Grassmannian G(k, V) of k-dimensional vector subspaces in V.