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I would like to know if there is anything known about the Hilbert-Chow morphism for the case of a singular curve. Due to Fogarty, for any number $n$ and any smooth projective variety $X$, there is a morphism $$ \alpha: Hilb^n(X)\to Sym^n(X),\; Z\mapsto \sum\limits_{p\in X}l_p(Z)\cdot p, $$ where $l_p(Z)$ denotes the length of the subscheme $Z$. In more geometric terms, this is saying that $Hilb^n(X)$ arises as blow-up of $Sym^2(X)$ along the diagonal.

What happens, if $n=2$ and $X=C$ a singular curve? Since I just want to get some intuition about what might happen for singular curves, feel free to only say something about $C$ being a nodal curve. I am also thankful to any kind of reference and/or suggestions for reading.

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For $n=2$ the fiber $\alpha^{-1}(2p)$ of the Hilbert-Chow morphism is just the projectivized tangent space $\mathbb{P}(T_pX)$.

So for example when $X=C$ is a curve and $p$ is a node point, the fiber is just a $\mathbb{P}^1$. Explicitly, if we pick coordinates so that the completed local ring at $p$ is $k[[x,y]]/(xy)$ then the ideals of colength $2$ vanishing at the origin are of the form $(ax+by,x^2,y^2)$ as $[a,b]$ varies in $\mathbb{P}^1$.

More generally, $\alpha^{-1}(np)$ inside $\mathrm{Hilb}^n(C)$ (sometimes called the punctual Hilbert scheme) has been studied quite extensively for a reduced curve singularity $p \in C$. The following list is far from exhaustive but here are some example references: this paper for unibranch curve singularities, here for planar curves and connections to knot theory, and here for general reduced curves.

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