If $Sigma$ is a smooth complex curve in a smooth projective surface $X$, then we can consider the homology class represented by $Sigma^{[n]} subset X^{[n]}$. $ $ Where, $X^{[n]}, Sigma^{[n]}$ stand for the Hilbert scheme of $n$-points on $X$ and $Sigma$, respectively. Is it possible to construct a homomorphism function $Phi_n: rm{H}_2(X) rightarrow H_w(X^{[n]})$, such that $[Sigma] mapsto [ Sigma^{[n]} ]$?
$ $ One has the following at ones disposal: we have the obvious quotient map $X^n rightarrow S^nX$ (where $S^nX$ is the symmetric product of $X$). Now, if $beta in H_2(X)$, then we can consider the image of $B := beta times cdots times beta$ in $H_{2n}(S^nX)$. If $beta $ can be represented by an algebraic curve, we can take the proper transform of $B$ under the Chow map $X^{[n]} rightarrow S^nX$. If $beta$ is not represented by such a curve, is there anything akin to proper transform that one can apply to $B$ to construct the desired homomorphism function $Phi_n$?
I am interested in studying the intersection theory between the classes $Phi_n(beta)$. Nakajima in his book "Lectures on Hilbert schemes of points on surfaces" states the following nice result. If $Sigma$ and $Sigma'$ are two smooth curves in $X$, then (page 102 of Nakajima's book):
$$sum_n z^n [Sigma^{[n]}] cdot [Sigma'^{[n]}] = (1+z)^{[Sigma] cdot [Sigma']}$$
Does anyone know if there are related results for singular curves?
As a side remark. the above formula is obvious if $Sigma$ and $Sigma'$ are two curves intersecting transversely. All it says is that of the set of $m = [Sigma]cdot [Sigma']$ points were it intersects, we choose $n$ of them (there are $binom{m}{n}$ of these guys, which is what the formula is giving). But the general proof of the formula is more intricate - one uses a representation of the Heisenberg group on the space $oplus_n H_*(X^{[n]})$ to derive it. This fancy shmancy approach is more helpful when computing things like the self intersection of $Sigma^{[n]}$ when $Sigma$ is a $(-1)$-curve in $X$. From it we get that $[Sigma^{[n]}] cdot[Sigma^{[n]}] = binom{-1}{n} = (-1)^n$
EDITED: In view of Nakajima's comment below, please replace function for homomorphism when reading the above question. Notice that, as stated in my comment below, the extension of the map $[Sigma] rightarrow [Sigma^{[n]}]$ should be a "nice" one.
EDITED (I am copying my hidden comments here since their maths don't display well)
I can explain my motivation. I am working with some moduli spaces of objects on a surface $X$ and out of them I get a homology class $V_n$ in $X^{[n]}$. In nice cases, one can show that these homology classes are $[Sigma^{[n]}]$, for some curve $Sigma subset X$. Or a sum of such classes. Using this classes $V_n$ I am trying to obtain a map $N : H_2(X) rightarrow mathbb{Z}$, defined by $N(beta) := V_n cdot Phi_n(beta)$. Such that, in the nice case when $V_n = [Sigma^{[n]}]$ and $beta = [Sigma']$, then $$N(beta) = [Sigma^{[n]}] cdot [Sigma'^{[n]}]$$
Then, my problem became what should be the definition of $Phi_n(beta)$, when $beta$ not represented by a curve. Presumably, we should be able to extend $Phi_n$ to some 2-classes that are not represented by curves since, by perturbing the complex structure, we could start seeing more curves than before. I don't know what should be $Phi_n(-2H)$. The best I could imagine is that it should satisfy the equation $$[Sigma^{[n]}] cdot Phi_n(-2H) = binom{Sigma cdot (-2H)}{n} $$ but I really don't know what it should be. Thanks a lot again!
EDIT I am now assume that the formula
$$alpha mapsto expleft( sum frac{z_i P_alpha[-i]}{(-1)^{i-1}i} right) cdot 1 $$
(the definition of the term $P_alpha[-i]$ can be found in Prof. Nakajima's book "Lectures on Hilbert schemes of points on surfaces" page 84), is well defined. By one of his results, $[Sigma] mapsto sum z^i [Sigma^{[n]}]$ (op. cit. page 99). If so, I presume this satisfy the posed question.
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