Lemma 44.3.5. Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$ such that the functors $\mathrm{Hilb}^ d_{X/S}$ are representable. The schemes $\underline{\mathrm{Hilb}}^ d_{X/S}$ are smooth over $S$ of relative dimension $d$.

**Proof.**
We have $\underline{\mathrm{Hilb}}^0_{X/S} = S$ and $\underline{\mathrm{Hilb}}^1_{X/S} = X$ thus the result is true for $d = 0, 1$. Assuming the result for $d$, we see that $\underline{\mathrm{Hilb}}^ d_{X/S} \times _ S X$ is smooth over $S$ (Morphisms, Lemma 29.34.5 and 29.34.4). Since $\underline{\mathrm{Hilb}}^ d_{X/S} \times _ S X \to \underline{\mathrm{Hilb}}^{d + 1}_{X/S}$ is finite locally free of degree $d + 1$ by Lemma 44.3.4 the result follows from Descent, Lemma 35.14.5. We omit the verification that the relative dimension is as claimed (you can do this by looking at fibres, or by keeping track of the dimensions in the argument above).
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (3)

Comment #4950 by SDIGR on

Comment #4951 by SDIGR on

Comment #5207 by Johan on