On some subvarieties of the Grassmann variety

Abstract

Let $\mathcal{S}$ be a Desarguesian $(t-1)$--spread of $PG(rt-1,q)$, $\Pi$ a $m$-dimensional subspace of $PG(rt-1,q)$ and $\Lambda$ the linear set consisting of the elements of $\mathcal{S}$ with non-empty intersection with $\Pi$. It is known that the Plücker embedding of the elements of $\mathcal{S}$ is a variety of $PG(r^t-1,q)$, say ${\mathcal{V}}_{rt}$. In this paper, we describe the image under the Plücker embedding of the elements of $\Lambda$ and we show that it is an $m$-dimensional algebraic variety, projection of a Veronese variety of dimension $m$ and degree $t$, and it is a suitable linear section of ${\mathcal{V}}_{rt}$.

Keywords: Grassmannian, linear set, Desarguesian spread, Schubert variety

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