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

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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 $P_{F}^{n}$.
Sia $F$ un campo e $Gr(i, F^{n})$ la Grassmanniana dei sottospazi $i$-dimensionali di $F^{n}$. Un’applicazione $f : Gr(i, F^{n}) \rightarrow Gr(j, F^{n})$ si dice «nesting» se $l \subset f(l)$ per ogni $l \in Gr(i, F^{n})$. Glover, Homer and Stong hanno dimostrato che non ci sono applicazioni continue «nesting» da $Gr(i, \mathbb{C}^{n}) \rightarrow Gr(j, \mathbb{C}^{n})$ a parte un piccolo numero di eccezioni. Dimostriamo un risultato analogo per applicazioni «nesting» algebriche $Gr(i, F^{n}) \rightarrow Gr(j, F^{n})$, nel caso in cui $F$ sia un campo algebricamente chiuso di caratteristica arbitraria. Per $i=1$ ciò implica una descrizione dei sottofibrati algebrici del fibrato tangente allo spazio proiettivo $P_{F}^{n}$. 
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     title = {Nesting maps of {Grassmannians}},
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De Concini, Corrado; Reichstein, Zinovy. Nesting maps of Grassmannians. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 2, pp. 109-118. http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_2_a4/

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