Embedding of symmetric square in Jacobian

Question

Let $C$ be a projective curve defined over a field $K$, and let $C^{(2)}$ and $J$ be its symmetric square and Jacobian, respectively.

There is a natural map $C^{(2)}\hookrightarrow J$, defined as follows: let $O\in J$ be a point of degree $2$, then map $P\in C^{(2)}\mapsto P - O$.

My question is: when is this map an embedding.

I understand the question is somehow related to the number of global sections of a degree 2 line bundle on $C$. I don't understand in what way. A reference would be appreciated as well.

Answer 1

Assume that the genus $g$ of $C$ is $\geq 2$, so that $J$ is at least $2$-dimensional. The map $\phi : C^{(2)} \to J$ is a closed inclusion if and only if $C$ is not hyperelliptic.

More generally, the following are equivalent:

(1) $\phi: C^{(r)} \to \text{Pic}^r(C)$ is a closed embedding.

(2) $\phi: C^{(r)} \to \text{Pic}^r(C)$ is injective on $\mathbb{C}$-points.

(3) $H^0(C, \mathcal{O}(x_1 + x_2 + \cdots + x_r)) = \mathbb{C}$ for all effective divisors $x_1 + x_2 + \cdots + x_r$ of degree $r$.

I'll do the easy implications and cite Milne for the hard one.

$(1) \implies (2)$ is by definition.

$(2) \Longleftrightarrow (3)$: We have $\phi(x_1, x_2, \cdots, x_r) = \phi(y_1, y_2, \ldots, y_r)$ if and only if the divisors $x_1+x_2+\cdots+x_r$ and $y_1+y_2 + \cdots + y_r$ are rationally equivalent. This occurs if and only if there is some $f \in H^0(C, \mathcal{O}(x_1 + x_2 + \cdots + x_r))$ with zeroes at $(y_1, y_2, \ldots, y_r)$ (with multiplicity). So asking that $\phi(x_1, x_2, \cdots, x_r) \neq \phi(y_1, y_2, \ldots, y_r)$ for all distinct points of $C^{(r)}$ is the same as asking that $H^0(C, \mathcal{O}(x_1 + x_2 + \cdots + x_r))$ have no nonconstant global sections for any $(x_1, x_2, \ldots, x_r) \in C^{(r)}$.

Now, suppose that both $(2)$ and $(3)$ hold (since we know that they are equivalent); we want to show $(1)$. Since $C^{(r)}$ is smooth and projective, map from $C^{(r)}$ will be a closed embedding if and only if it is injective and injective on tangent spaces, and $(2)$ shows that the map is injective. It remains to check injectivity on tangent spaces.

So, given $x_1$, $x_2$, ..., $x_r \in C$, we want to show that $$D\phi : T_{(x_1, x_2, \ldots, x_r)} C^{(r)} \to T_{\phi(x_1, x_2, \ldots, x_r)} J \qquad (\ast)$$ is injective. Theorem 5.1 of Milne's notes on the Jacobian shows that the kernel of $(\ast)$ has dimension $\dim H^0(C, \mathcal{O}(x_1+x_2+\cdots+x_r))-1$. So, using $(3)$, we are done.