Nefness of $h-e$ in the blowup of $\mathbb{P}^n$

Question

Let $S$ be the blow up of $\mathbb{P}^n$ in a point $P$. Let $h$ be the class of the pullback of an hyperplane of $\mathbb{P}^n$ and $e$ the class of the exceptional divisor. Why is the divisor $l=h-e$ nef? Thank you very much!

Answer 1

Not every effective divisor is nef. By definition a divisor is nef if it intersects every curve non-negativelly. For instance the exceptional divisor $E$ is effective but not nef as it intersects any line contained in it negatively.

To see that $\tilde H$ is nef one can use Emerton's argument to show that the linear system $|\tilde H|$ is free from base points since it contains all the strict transforms of hyperplanes through $P$. So given a curve $C \subset S$ we can choose among these strict transforms one which does not contain $C$ to show that $\tilde H \cdot C \ge 0$.

Answer 2

We can choose a representative of $h$ which is the preimage of a hyperplane $H$ passing through $P$. This preimage is then equal to $\tilde{H} + E$, where $\tilde{H}$ is the proper transform of $H$ and $E$ is the exceptional divisor. Thus $\tilde{H}$ is a representative for $h - e,$ and is an effective divisor.

Answer 3

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

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, let $H$ be a hyperplane passing through $P$ but not containing $C$. Then $\tilde{H}$ does not contain $C$ and hence $(h-e)C=\tilde{D}C\geq 0$. If $C$ is contained in $E$, then $h\cdot C =0$. But $-E\cdot C =-\deg N_{E/X}|C\geq 1$, where $N_{E/X}$ is the normal bundle of the exceptional divisor in the blowing up $X$. Therefore $h-e$ is nef.