## 44.3 Moduli of divisors on smooth curves

For a smooth morphism $X \to S$ of relative dimension $1$ the functor $\mathrm{Hilb}^ d_{X/S}$ parametrizes relative effective Cartier divisors as defined in Divisors, Section 31.18.

Lemma 44.3.1. Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D \subset X$ be a closed subscheme. Consider the following conditions

1. $D \to S$ is finite locally free,

2. $D$ is a relative effective Cartier divisor on $X/S$,

3. $D \to S$ is locally quasi-finite, flat, and locally of finite presentation, and

4. $D \to S$ is locally quasi-finite and flat.

We always have the implications

$(1) \Rightarrow (2) \Leftrightarrow (3) \Rightarrow (4)$

If $S$ is locally Noetherian, then the last arrow is an if and only if. If $X \to S$ is proper (and $S$ arbitrary), then the first arrow is an if and only if.

Proof. Equivalence of (2) and (3). This follows from Divisors, Lemma 31.18.9 if we can show the equivalence of (2) and (3) when $S$ is the spectrum of a field $k$. Let $x \in X$ be a closed point. As $X$ is smooth of relative dimension $1$ over $k$ and we see that $\mathcal{O}_{X, x}$ is a regular local ring of dimension $1$ (see Varieties, Lemma 33.25.3). Thus $\mathcal{O}_{X, x}$ is a discrete valuation ring (Algebra, Lemma 10.119.7) and hence a PID. It follows that every sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ which is nonvanishing at all the generic points of $X$ is invertible (Divisors, Lemma 31.15.2). In other words, every closed subscheme of $X$ which does not contain a generic point is an effective Cartier divisor. It follows that (2) and (3) are equivalent.

If $S$ is Noetherian, then any locally quasi-finite morphism $D \to S$ is locally of finite presentation (Morphisms, Lemma 29.21.9), whence (3) is equivalent to (4).

If $X \to S$ is proper (and $S$ is arbitrary), then $D \to S$ is proper as well. Since a proper locally quasi-finite morphism is finite (More on Morphisms, Lemma 37.40.1) and a finite, flat, and finitely presented morphism is finite locally free (Morphisms, Lemma 29.48.2), we see that (1) is equivalent to (2). $\square$

Lemma 44.3.2. Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. Then $D_1 + D_2$ is finite locally free of degree $d_1 + d_2$ over $S$.

Proof. By Lemma 44.3.1 we see that $D_1$ and $D_2$ are relative effective Cartier divisors on $X/S$. Thus $D = D_1 + D_2$ is a relative effective Cartier divisor on $X/S$ by Divisors, Lemma 31.18.3. Hence $D \to S$ is locally quasi-finite, flat, and locally of finite presentation by Lemma 44.3.1. Applying Morphisms, Lemma 29.41.11 the surjective integral morphism $D_1 \amalg D_2 \to D$ we find that $D \to S$ is separated. Then Morphisms, Lemma 29.41.9 implies that $D \to S$ is proper. This implies that $D \to S$ is finite (More on Morphisms, Lemma 37.40.1) and in turn we see that $D \to S$ is finite locally free (Morphisms, Lemma 29.48.2). Thus it suffice to show that the degree of $D \to S$ is $d_1 + d_2$. To do this we may base change to a fibre of $X \to S$, hence we may assume that $S = \mathop{\mathrm{Spec}}(k)$ for some field $k$. In this case, there exists a finite set of closed points $x_1, \ldots , x_ n \in X$ such that $D_1$ and $D_2$ are supported on $\{ x_1, \ldots , x_ n\}$. In fact, there are nonzerodivisors $f_{i, j} \in \mathcal{O}_{X, x_ i}$ such that

$D_1 = \coprod \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}/(f_{i, 1})) \quad \text{and}\quad D_2 = \coprod \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}/(f_{i, 2}))$

Then we see that

$D = \coprod \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}/(f_{i, 1}f_{i, 2}))$

From this one sees easily that $D$ has degree $d_1 + d_2$ over $k$ (if need be, use Algebra, Lemma 10.121.1). $\square$

Lemma 44.3.3. Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. If $D_1 \subset D_2$ (as closed subschemes) then there is a closed subscheme $D \subset X$ finite locally free of degree $d_2 - d_1$ over $S$ such that $D_2 = D_1 + D$.

Proof. This proof is almost exactly the same as the proof of Lemma 44.3.2. By Lemma 44.3.1 we see that $D_1$ and $D_2$ are relative effective Cartier divisors on $X/S$. By Divisors, Lemma 31.18.4 there is a relative effective Cartier divisor $D \subset X$ such that $D_2 = D_1 + D$. Hence $D \to S$ is locally quasi-finite, flat, and locally of finite presentation by Lemma 44.3.1. Since $D$ is a closed subscheme of $D_2$, we see that $D \to S$ is finite. It follows that $D \to S$ is finite locally free (Morphisms, Lemma 29.48.2). Thus it suffice to show that the degree of $D \to S$ is $d_2 - d_1$. This follows from Lemma 44.3.2. $\square$

Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. By Lemma 44.3.1 for a scheme $T$ over $S$ and $D \in \mathrm{Hilb}^ d_{X/S}(T)$, we can view $D$ as a relative effective Cartier divisor on $X_ T/T$ such that $D \to T$ is finite locally free of degree $d$. Hence, by Lemma 44.3.2 we obtain a transformation of functors

$\mathrm{Hilb}^{d_1}_{X/S} \times \mathrm{Hilb}^{d_2}_{X/S} \longrightarrow \mathrm{Hilb}^{d_1 + d_2}_{X/S},\quad (D_1, D_2) \longmapsto D_1 + D_2$

If $\mathrm{Hilb}^ d_{X/S}$ is representable for all degrees $d$, then this transformation of functors corresponds to a morphism of schemes

$\underline{\mathrm{Hilb}}^{d_1}_{X/S} \times _ S \underline{\mathrm{Hilb}}^{d_2}_{X/S} \longrightarrow \underline{\mathrm{Hilb}}^{d_1 + d_2}_{X/S}$

over $S$. Observe that $\underline{\mathrm{Hilb}}^0_{X/S} = S$ and $\underline{\mathrm{Hilb}}^1_{X/S} = X$. A special case of the morphism above is the morphism

$\underline{\mathrm{Hilb}}^ d_{X/S} \times _ S X \longrightarrow \underline{\mathrm{Hilb}}^{d + 1}_{X/S},\quad (D, x) \longmapsto D + x$

Lemma 44.3.4. 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 morphism $\underline{\mathrm{Hilb}}^ d_{X/S} \times _ S X \to \underline{\mathrm{Hilb}}^{d + 1}_{X/S}$ is finite locally free of degree $d + 1$.

Proof. Let $D_{univ} \subset X \times _ S \underline{\mathrm{Hilb}}^{d + 1}_{X/S}$ be the universal object. There is a commutative diagram

$\xymatrix{ \underline{\mathrm{Hilb}}^ d_{X/S} \times _ S X \ar[rr] \ar[rd] & & D_{univ} \ar[ld] \ar@{^{(}->}[r] & \underline{\mathrm{Hilb}}^{d + 1}_{X/S} \times _ S X \\ & \underline{\mathrm{Hilb}}^{d + 1}_{X/S} }$

where the top horizontal arrow maps $(D', x)$ to $(D' + x, x)$. We claim this morphism is an isomorphism which certainly proves the lemma. Namely, given a scheme $T$ over $S$, a $T$-valued point $\xi$ of $D_{univ}$ is given by a pair $\xi = (D, x)$ where $D \subset X_ T$ is a closed subscheme finite locally free of degree $d + 1$ over $T$ and $x : T \to X$ is a morphism whose graph $x : T \to X_ T$ factors through $D$. Then by Lemma 44.3.3 we can write $D = D' + x$ for some $D' \subset X_ T$ finite locally free of degree $d$ over $T$. Sending $\xi = (D, x)$ to the pair $(D', x)$ is the desired inverse. $\square$

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 35.11.5. 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). $\square$

We collect all the information obtained sofar in the case of a proper smooth curve over a field.

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. $\square$

Remark 44.3.7. Let $X$ be a geometrically irreducible smooth proper curve over a field $k$ as in Proposition 44.3.6. Let $d \geq 0$. The universal closed object is a relatively effective divisor

$D_{univ} \subset \underline{\mathrm{Hilb}}^{d + 1}_{X/k} \times _ k X$

over $\underline{\mathrm{Hilb}}^{d + 1}_{X/k}$ by Lemma 44.3.1. In fact, $D_{univ}$ is isomorphic as a scheme to $\underline{\mathrm{Hilb}}^ d_{X/k} \times _ k X$, see proof of Lemma 44.3.4. In particular, $D_{univ}$ is an effective Cartier divisor and we obtain an invertible module $\mathcal{O}(D_{univ})$. If $[D] \in \underline{\mathrm{Hilb}}^{d + 1}_{X/k}$ denotes the $k$-rational point corresponding to the effective Cartier divisor $D \subset X$ of degree $d + 1$, then the restriction of $\mathcal{O}(D_{univ})$ to the fibre $[D] \times X$ is $\mathcal{O}_ X(D)$.

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