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.



The canonical divisor of the Hilbert scheme $Hilb^n P^2$?
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.


