The Stacks project

Proposition 44.3.6. Let $X$ be a geometrically irreducible smooth proper curve over a field $k$.

  1. The functors $\mathrm{Hilb}^ d_{X/k}$ are representable by smooth proper varieties $\underline{\mathrm{Hilb}}^ d_{X/k}$ of dimension $d$ over $k$.

  2. For a field extension $k'/k$ the $k'$-rational points of $\underline{\mathrm{Hilb}}^ d_{X/k}$ are in $1$-to-$1$ bijection with effective Cartier divisors of degree $d$ on $X_{k'}$.

  3. For $d_1, d_2 \geq 0$ there is a morphism

    \[ \underline{\mathrm{Hilb}}^{d_1}_{X/k} \times _ k \underline{\mathrm{Hilb}}^{d_2}_{X/k} \longrightarrow \underline{\mathrm{Hilb}}^{d_1 + d_2}_{X/k} \]

    which is finite locally free of degree ${d_1 + d_2 \choose d_1}$.

Proof. The functors $\mathrm{Hilb}^ d_{X/k}$ are representable by Proposition 44.2.6 (see also Remark 44.2.7) and the fact that $X$ is projective (Varieties, Lemma 33.42.4). The schemes $\underline{\mathrm{Hilb}}^ d_{X/k}$ are separated over $k$ by Lemma 44.2.4. The schemes $\underline{\mathrm{Hilb}}^ d_{X/k}$ are smooth over $k$ by Lemma 44.3.5. Starting with $X = \underline{\mathrm{Hilb}}^1_{X/k}$, the morphisms of Lemma 44.3.4, and induction we find a morphism

\[ X^ d = X \times _ k X \times _ k \ldots \times _ k X \longrightarrow \underline{\mathrm{Hilb}}^ d_{X/k},\quad (x_1, \ldots , x_ d) \longrightarrow x_1 + \ldots + x_ d \]

which is finite locally free of degree $d!$. Since $X$ is proper over $k$, so is $X^ d$, hence $\underline{\mathrm{Hilb}}^ d_{X/k}$ is proper over $k$ by Morphisms, Lemma 29.41.9. Since $X$ is geometrically irreducible over $k$, the product $X^ d$ is irreducible (Varieties, Lemma 33.8.4) hence the image is irreducible (in fact geometrically irreducible). This proves (1). Part (2) follows from the definitions. Part (3) follows from the commutative diagram

\[ \xymatrix{ X^{d_1} \times _ k X^{d_2} \ar[d] \ar@{=}[r] & X^{d_1 + d_2} \ar[d] \\ \underline{\mathrm{Hilb}}^{d_1}_{X/k} \times _ k \underline{\mathrm{Hilb}}^{d_2}_{X/k} \ar[r] & \underline{\mathrm{Hilb}}^{d_1 + d_2}_{X/k} } \]

and multiplicativity of degrees of finite locally free morphisms. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B9I. Beware of the difference between the letter 'O' and the digit '0'.