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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 $$\omega_{X^{[n]}}=\omega_{X}^{[n]}.$$ $I think this is in GĂ¶ttsche's book on Hilbert schemes of points. 1 There is an easy formula for the canonical divisor on the Hilbert scheme of$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 $$\omega_{X^{[n]}}=\omega_{X}^{[n]}.$$