# Fibers of the Hilbert-Chow morphism vs local punctual Hilbert schemes

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 $$\varphi_{n,X}^{-1}\left( \sum n_i x_i \right)$$?

• 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 $H_{n_i,X,x_i}$. Apr 26, 2019 at 8:47
• 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 Apr 30, 2019 at 10:44
• 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 $n_i$. Apr 30, 2019 at 12:01