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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 $$\sum z^n [S^n \alpha \Sigma] = mapsto exp\left( \sum \frac{z_i P_\Sigma[i]}{(-1)^{i-1}iP_\alpha[-i]}{(-1)^{i-1}i} \right) \cdot 1$$ that (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 9984), is well definedif we replace $\Sigma$ by any 2-homology class . By one of his results, $\alpha$. [\Sigma] \mapsto \sum z^i [\Sigma^{[n]}]$(op. cit. page 99). If so, I presume this satisfy the posed question. 5 added 372 characters in body If$\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 $$\sum z^n [S^n \Sigma] = exp\left( \sum \frac{z_i P_\Sigma[i]}{(-1)^{i-1}i} \right) \cdot 1$$ that can be found in Prof. Nakajima's book "Lectures on Hilbert schemes of points on surfaces" page 99, is well defined if we replace$\Sigma$by any 2-homology class$\alpha$. If so, I presume this satisfy the posed question. 4 Maths don't display in comment section when comments get hidden, i copied them here. 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!

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