Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hey everyone,

I was wondering if anyone knows what the canonical divisor of the Hilbert scheme $Hilb^n P^2$ is --$Hilb^n P^2$ is the Hilbert scheme of degree-n zero dimensional subschemes of the projective plane $P^2$. Any references?

Many thanks in advance.

share|improve this question
add comment

2 Answers 2

up vote 6 down vote accepted

There is an easy formula for the canonical divisor on the Hilbert scheme of $n$ poin ts on any any smooth projective surface $X$. Let's first fix some notations. Denote by $X^{n}$ the $n-$fold product with projections $pr_i X^{n}\to X$. We can consider line bundles of the form $$ L^{[n]}=pr_1^* L \otimes\cdots \otimes pr_n^* L $$It is not hard to show that this decends to a line bundle on the symmetric product ${X}^{[n]}$ and this defines a homomorphism $Pic(X)\to Pic X^{[n]}$. In this notation, the canonical line bundle is given by $$ \omega_{X^{[n]}}=\omega_{X}^{[n]}. $$I think this is in Göttsche's book on Hilbert schemes of points.

share|improve this answer
    
Thank you. It was very helpful. –  Turkelli Feb 18 '11 at 18:37
add comment

$n=1$ already tells you that the anticanonical divisor is going to be nicer than the canonical, in that it's effective. There, the divisor given by the three coordinate lines is anticanonical.

Next step, look at the Chow variety of $n$ points in $P^2$, with an anticanonical given by "some point is on some coordinate line".

Then use the fact that the morphism from Hilb to Chow is crepant, to say that we can pull the anticanonical back. So: the divisor given by "some point is on some coordinate line" is again anticanonical up on the Hilbert scheme.

share|improve this answer
    
Oh, I see, thank you. In fact, I care about the anti-canonical divisor (rather than the canonical) as I am interested in Batyrev-Manin conjectures for Hilbert schemes. I wonder if the effective cone of the Hilbert schemes is known? –  Turkelli Feb 19 '11 at 21:32
add comment

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.