ag.algebraic geometry - Nefness of $h-e$ in the blowup of $mathbb{P}^n$

As a complement to JVP's answer, here is a direct proof to that $tilde{H}cdot Cgeq 0$.

Note that nefness is numerically invariant. To check the nefness of $h-e$, we only need to show that for any irreducible curve $C$ in the blowing-up, the intersection number $(h-e)cdot C$ is nonnegative. If $C$ is not contained in $e$. then the image of $C$, denoted by $D$, is still a curve. In this case, by projection formular, $(h-e)cdot C=Hcdot Dgeq 0$, where $H$ is any hyperplane in $mathbb{P}^n$. In the case $C$ is contained in $e$, $hcdot C =0$ however $-ecdot C =-deg N_{e/X}|C=1$, where $N_{e/X}$ is the normal bundle of the exceptional divisor in the blowing up $X$. Therefore $h-e$ is nef.