MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

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 $\tilde H \cdot C \ge 0$.

share|cite|improve this answer

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.

share|cite|improve this answer
    
Not every effective divisor is nef. – Jorge Vitório Pereira Jan 29 '10 at 22:37
    
Thanks. To complete the argument, see jvp's answer. – Emerton Jan 30 '10 at 2:08

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, by projection formular, $(h-e)\cdot C=H\cdot D\geq 0$, where $H$ is any hyperplane in $\mathbb{P}^n$. In the case $C$ is contained in $e$, $h\cdot C =0$ however $-e\cdot C =-\deg N_{e/X}|C=1$, where $N_{e/X}$ is the normal bundle of the exceptional divisor in the blowing up $X$. Therefore $h-e$ is nef.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.