Embeddings of Grassmann spaces

Abstract

In this paper we show a construction, inductive onn, of the Grassmannian$\mathcal{G}_{n,1,F}$ representing the lines of the projective spaceP _n,F :$\mathcal{G}_{n,1,F}$ is the union of those planes, any of which is obtained by joining a line of a certain (n−1)-flat S _n−1 with the corresponding point on$\mathcal{G}_{n - 1,1,F}$ via the Plücker mapping; it is assumed that the$\left( {\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right) - 1} \right)$-flat spanned by this$\mathcal{G}_{n - 1,1,F}$ is skew with S _n−1.

Every embedded Grassmann space can be obtained from a Grassmannian by projecting it on a subspaceS from another one, sayS′ (the vertex of the projection). This is a consequence of the description of all projective embeddings of the Grassmann spaces given by HAVLICEK [4] and WELLS [9]. When the vertexS′ is not empty, we say that the embedded Grassmann space is a projected Grassmannian. Since examples of projected Grassmannians do exist, the embedded$\mathcal{G}_{n,h,F}$'s cannot, in general, be projectively characterized by using only their intrinsic incidence properties. We show that for any fieldF, the projection is impossible exactly forn=3,4. In every other case the dimension of the vertex can depend onF.