Introduction

Given a vector space $V_n$ of dimension $n$ over an algebraically closed field $K$ , the Grassmannian $Gr(r, V_n)$ is the collection of all $r$ -dimensional subspaces of $V_n$ with the quotient topology. It is immediate that $Gr(1, \mathbb{C}^n) = \mathbb{C}\mathbb{P}^{n-1}$ , hence we may think of it as a generalization of the usual projective spaces. In fact $Gr(r, V_n)$ is itself a projective variety (see the Plücker imbedding).

Tag: Zariski

The Grassmannian (1)

Introduction