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Let $X$ be a curve over a scheme $k$: Let $H_{n,X}$ be the punctual scheme of $X$ parametrizing finite subschemes of degree $n$, and le $\varphi_{n,X}: H_{n,X} \rightarrow X^{(n)}$ be the Hilbert-Chow morphism. For a closed point $x \in X$, be the local punctual Hilbert scheme $H_{n,X,x} = \varphi_{n,X}^{-1}(nx)$ is known in many cases (for example, curves with nodal singularities).

Suppose that $x_1, \dots, x_r$ are distinct points of $X$. Is it possible to deduce something on the geometry of $\varphi_{n,X}^{-1}\left( \sum n_i x_i \right)$ from the geometry of the $H_{n,X,x_i}$'s?

For example, on a node $x$, any cluster $Z \in H_{n,X,x}$ which is not defined by a single local equation is a limit of subschemes defined by single equations (Example 2.1 in Bertin's The punctual Hilbert scheme: an Introduction). If $x_1, \dots, x_r$ are nodes, does a similar fact hold also for $\varphi_{n,X}^{-1}\left( \sum n_i x_i \right)$?

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    $\begingroup$ Since the points are distinct, the fiber of the Hilbert-Chow morphism over $\sum n_i x_i$ is isomorphic to the product of the punctual Hilbert schemes $H_{n_i,X,x_i}$. $\endgroup$
    – Daniele A
    Apr 26, 2019 at 8:47
  • $\begingroup$ Is it trivial or do you have any reference? I feel that this should be true, but it seems to me strange that e.g. Bertin does not show it $\endgroup$
    – Raffaele C
    Apr 30, 2019 at 10:44
  • $\begingroup$ Well, it is true from the definition: $\xi\subseteq X$ is a scheme in the fiber precisely when $\xi$ is supported at the points $x_i$ and has length $n_i$ at each point. Then $\xi$ is a disjoint union of subsechemes $\xi_i$, each supported at $x_i$ and of length $n_i$. $\endgroup$
    – Daniele A
    Apr 30, 2019 at 12:01

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