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 $PGL\left(n\right)$.

In this paper, we give a numerical characterization of nef arithmetic $ℝ$-Cartier divisors of $C^0$-type on an arithmetic surface. Namely an arithmetic $ℝ$-Cartier divisor $\overline{D}$ of $C^0$-type is nef if and only if $\overline{D}$ is pseudo-effective and $\widehat{\mathrm{deg}}\left({\overline{D}}^2\right)=\widehat{\mathrm{vol}}\left(\overline{D}\right)$.