Let $F$ be a field and $Gr\left(i,F^n\right)$ be the Grassmannian of $i$-dimensional linear subspaces of $F^n$. A map $f:Gr\left(i,F^n\right)\to Gr\left(j,F^n\right)$ is called nesting if $l\subset f\left(l\right)$ for every $l\in Gr\left(i,F^n\right)$. Glover, Homer and Stong showed that there are no continuous nesting maps $Gr\left(i,{\mathbb{C}}^n\right)\to Gr\left(j,{\mathbb{C}}^n\right)$ except for a few obvious ones. We prove a similar result for algebraic nesting maps $Gr\left(i,F^n\right)\to Gr\left(j,F^n\right)$, 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 $P_F^n$.

Inspired by Manin’s approach towards a geometric interpretation of Arakelov theory at infinity, we interpret in this paper non-Archimedean local intersection numbers of linear cycles in ${\mathbb{P}}^{n-1}$ with the combinatorial geometry of the Bruhat-Tits building associated to $PGL\left(n\right)$.