The Stacks project

Lemma 44.6.7. Let $k$ be a separably closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$.

  1. $\underline{\mathrm{Pic}}_{X/k}$ is a disjoint union of $g$-dimensional smooth proper varieties $\underline{\mathrm{Pic}}^ d_{X/k}$,

  2. $k$-points of $\underline{\mathrm{Pic}}^ d_{X/k}$ correspond to invertible $\mathcal{O}_ X$-modules of degree $d$,

  3. $\underline{\mathrm{Pic}}^0_{X/k}$ is an open and closed subgroup scheme,

  4. for $d \geq 0$ there is a canonical morphism $\gamma _ d : \underline{\mathrm{Hilb}}^ d_{X/k} \to \underline{\mathrm{Pic}}^ d_{X/k}$

  5. the morphisms $\gamma _ d$ are surjective for $d \geq g$ and smooth for $d \geq 2g - 1$,

  6. the morphism $\underline{\mathrm{Hilb}}^ g_{X/k} \to \underline{\mathrm{Pic}}^ g_{X/k}$ is birational.

Proof. Pick a $k$-rational point $\sigma $ of $X$. Recall that $\mathrm{Pic}_{X/k}$ is isomorphic to the functor $\mathrm{Pic}_{X/k, \sigma }$. By Derived Categories of Schemes, Lemma 36.32.2 for every $d \in \mathbf{Z}$ there is an open subfunctor

\[ \mathrm{Pic}^ d_{X/k, \sigma } \subset \mathrm{Pic}_{X/k, \sigma } \]

whose value on a scheme $T$ over $k$ consists of those $\mathcal{L} \in \mathrm{Pic}_{X/k, \sigma }(T)$ such that $\chi (X_ t, \mathcal{L}_ t) = d + 1 - g$ and moreover we have

\[ \mathrm{Pic}_{X/k, \sigma } = \coprod \nolimits _{d \in \mathbf{Z}} \mathrm{Pic}^ d_{X/k, \sigma } \]

as fppf sheaves. It follows that the scheme $\underline{\mathrm{Pic}}_{X/k}$ (which exists by Proposition 44.6.6) has a corresponding decomposition

\[ \underline{\mathrm{Pic}}_{X/k, \sigma } = \coprod \nolimits _{d \in \mathbf{Z}} \underline{\mathrm{Pic}}^ d_{X/k, \sigma } \]

where the points of $\underline{\mathrm{Pic}}^ d_{X/k, \sigma }$ correspond to isomorphism classes of invertible modules of degree $d$ on $X$.

Fix $d \geq 0$. There is a morphism

\[ \gamma _ d : \underline{\mathrm{Hilb}}^ d_{X/k} \longrightarrow \underline{\mathrm{Pic}}^ d_{X/k} \]

coming from the invertible sheaf $\mathcal{O}(D_{univ})$ on $\underline{\mathrm{Hilb}}^ d_{X/k} \times _ k X$ (Remark 44.3.7) by the Yoneda lemma (Categories, Lemma 4.3.5). Our proof of the representability of the Picard functor of $X/k$ in Proposition 44.6.6 and Lemma 44.6.4 shows that $\gamma _ g$ induces an open immersion on a nonempty open of $\underline{\mathrm{Hilb}}^ g_{X/k}$. Moreover, the proof shows that the translates of this open by $k$-rational points of the group scheme $\underline{\mathrm{Pic}}_{X/k}$ define an open covering. Since $\underline{\mathrm{Hilb}}^ g_{X/K}$ is smooth of dimension $g$ (Proposition 44.3.6) over $k$, we conclude that the group scheme $\underline{\mathrm{Pic}}_{X/k}$ is smooth of dimension $g$ over $k$.

By Groupoids, Lemma 39.7.3 we see that $\underline{\mathrm{Pic}}_{X/k}$ is separated. Hence, for every $d \geq 0$, the image of $\gamma _ d$ is a proper variety over $k$ (Morphisms, Lemma 29.41.10).

Let $d \geq g$. Then for any field extension $K/k$ and any invertible $\mathcal{O}_{X_ K}$-module $\mathcal{L}$ of degree $d$, we see that $\chi (X_ K, \mathcal{L}) = d + 1 - g > 0$. Hence $\mathcal{L}$ has a nonzero section and we conclude that $\mathcal{L} = \mathcal{O}_{X_ K}(D)$ for some divisor $D \subset X_ K$ of degree $d$. It follows that $\gamma _ d$ is surjective.

Combining the facts mentioned above we see that $\underline{\mathrm{Pic}}^ d_{X/k}$ is proper for $d \geq g$. This finishes the proof of (2) because now we see that $\underline{\mathrm{Pic}}^ d_{X/k}$ is proper for $d \geq g$ but then all $\underline{\mathrm{Pic}}^ d_{X/k}$ are proper by translation.

It remains to prove that $\gamma _ d$ is smooth for $d \geq 2g - 1$. Consider an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ of degree $d$. Then the fibre of the point corresponding to $\mathcal{L}$ is

\[ Z = \{ D \subset X \mid \mathcal{O}_ X(D) \cong \mathcal{L}\} \subset \underline{\mathrm{Hilb}}^ d_{X/k} \]

with its natural scheme structure. Since any isomorphism $\mathcal{O}_ X(D) \to \mathcal{L}$ is well defined up to multiplying by a nonzero scalar, we see that the canonical section $1 \in \mathcal{O}_ X(D)$ is mapped to a section $s \in \Gamma (X, \mathcal{L})$ well defined up to multiplication by a nonzero scalar. In this way we obtain a morphism

\[ Z \longrightarrow \text{Proj}(\text{Sym}(\Gamma (X, \mathcal{L})^*)) \]

(dual because of our conventions). This morphism is an isomorphism, because given an section of $\mathcal{L}$ we can take the associated effective Cartier divisor, in other words we can construct an inverse of the displayed morphism; we omit the precise formulation and proof. Since $\dim H^0(X, \mathcal{L}) = d + 1 - g$ for every $\mathcal{L}$ of degree $d \geq 2g - 1$ by Varieties, Lemma 33.43.17 we see that $\text{Proj}(\text{Sym}(\Gamma (X, \mathcal{L})^*)) \cong \mathbf{P}^{d - g}_ k$. We conclude that $\dim (Z) = \dim (\mathbf{P}^{d - g}_ k) = d - g$. We conclude that the fibres of the morphism $\gamma _ d$ all have dimension equal to the difference of the dimensions of $\underline{\mathrm{Hilb}}^ d_{X/k}$ and $\underline{\mathrm{Pic}}^ d_{X/k}$. It follows that $\gamma _ d$ is flat, see Algebra, Lemma 10.128.1. As moreover the fibres are smooth, we conclude that $\gamma _ d$ is smooth by Morphisms, Lemma 29.34.3. $\square$


Comments (2)

Comment #1880 by Keenan Kidwell on

In the third line of the final text block of the proof (below the displayed equation ), "inverset" should be "inverse."


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 0BA0. Beware of the difference between the letter 'O' and the digit '0'.