Plücker embedding
In mathematics, the Plücker map embeds the Grassmannian G r ( k , V ) {\displaystyle \mathrm {Gr} (k,V)} , whose elements are k-dimensional subspaces of an n-dimensional vector space V, either real or complex, in a projective space, thereby realizing it as a projective algebraic variety. More precisely, the Plücker map embeds G r ( k , V ) {\displaystyle \mathrm {Gr} (k,V)} into the projectivization P ( ⋀ k V ) {\displaystyle \mathbb {P} ({\textstyle \bigwedge }^{k}V)} of the k {\displaystyle k} -th exterior power of V {\displaystyle V} .