Nesting maps of Grassmannians

De Concini, Corrado ; Reichstein, Zinovy

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 2, pp. 109-118

Résumé

Let $F$ be a field and $Gr(i, F^{n})$ be the Grassmannian of $i$-dimensional linear subspaces of $F^{n}$. A map $f : Gr(i, F^{n}) \rightarrow Gr(j, F^{n})$ is called nesting if $l \subset f(l)$ for every $l \in Gr(i, F^{n})$. Glover, Homer and Stong showed that there are no continuous nesting maps $Gr(i, \mathbb{C}^{n}) \rightarrow Gr(j, \mathbb{C}^{n})$ except for a few obvious ones. We prove a similar result for algebraic nesting maps $Gr(i, F^{n}) \rightarrow Gr(j, F^{n})$, where $F$ is an algebraically closed field of arbitrary characteristic. For $i=1$ this yields a description of the algebraic sub-bundles of the tangent bundle to the projective space $P_{F}^{n}$.