Canonical divisor on the symmetric product of a hyperelliptic curve Let $C$ be a hyperelliptic curve of genus $g$ and let $S = C^{(2)}$ denote the symmetric square of $C$.  Let $\nabla$ be the divisor on $C^2$ defined by $\{(P, \overline{P}) \mid P \in C\}$ where $\overline{P}$ denotes the image of $P$ under the hyperelliptic involution.  Finally let $\nabla_S$ be the push-forward of $\nabla$ under the quotient map $C^2 \to S$.
If $g = 2$, then $S$ is the blowup of the Jacobian $J_C$ of $C$ at the origin with exceptional divisor $\nabla_S$.  So, by results from Hartshorne Section V.3 on the relationships between blowups and the intersection pairing, we immediately conclude that $\nabla_S$ is a canonical divisor of $S$ and it has self-intersection $-1$.  I would like to know how to do these calculations without the crutch of using blowups, and hence, with any luck, obtain the analogous results for when $g>2$. My question is therefore

How do I calculate a canonical divisor, and its self-intersection number, of the symmetric square of a hyperelliptic curve of genus $g>2$?

 A: Let $\pi: C^2\to S$ be the canonical quotient morphism. It is étale outside of the diagonal $\Delta$ of $C^2$. Let $K$ be the base field. The canonical map  of differentials
$$ \pi^*\Omega_{S/K}^1\to \Omega^1_{C^2/K}$$ 
is then an isomorphism outside of $\Delta$, and induces an injective homomorphism $\pi^*\omega_{S/K}\to \omega_{C^2/K}$ of canonical sheaves. So $\pi^*\omega_{S/K}=\omega_{C^2/K}(-D)$ for some effective divisor $D$ on $C^2$, with support in $\Delta$, hence $D=r\Delta$ for some integer $r\ge 0$. The multiplicity $r$ can be computed locally for the Zariski topology, and even for étale topology on $C$. So we can work with Spec ($K[x]$) and find $r=1$:
$$ \pi^*\omega_{S/K}=\omega_{C^2/K}(-\Delta).$$
This should be enough to describe a canonical divisor on $S$ interms of the pushforward of a canonical divisor of $C^2$ and of $\pi(\Delta)$. The self-intersection should be easily computed with the projection formula. Otherwise ask for more details.
Edit Computation of the multiplicity $r$. Let $\xi$ be the generic point of $\Delta$ and $\eta=\pi(\xi)$. Then 
$$ \omega_{S/K, \xi}\otimes O_{C^2,\xi}=(\pi^*\omega_{S/K})_{\xi}=\omega_{C^2/K}(-r\Delta)_{\xi}=\omega_{C^2/K,\xi}(-r\Delta).$$
This explains why $r$ can be computed Zariski locally. 
Let $U$ be a dense open subset of $C$. Then one can compute $r$ on $U^2\to U^{(2)}$. If we can write $U$ as an étale cover $U\to V\subseteq \mathbb A^1_K$, then the map 
$$ \pi^*\Omega_{U^{(2)}/K}^1\to \Omega^1_{U^2/K}$$ 
is just the pull-back of the map 
$$ \pi^*\Omega_{V^{(2)}/K}^1\to \Omega^1_{V^2/K}.$$ 
If you take a local basis $dx, dy$ for $\Omega^1_{V^2/K}$, then $d(x+y), d(xy)$ is a local basis for $\Omega^1_{V^{(2)}/K}$, and their pull-backs to $U^2$ (resp. $U^{(2)}$) are local bases, and $r$ can be computed with these local bases. Now $\omega_{V^2/K}$ is generated by $dx\wedge dy$, and $\omega_{V^{(2)}/K}$ is generated by $d(x+y)\wedge d(xy)$ who image in $\omega_{V^2/K}$ is $(x-y)(dx\wedge dy)$. As $x-y$ generates locally the ideal of $\Delta$, we see that $r=1$.  
A: The self intersection of the diagonal of a curve $C$ in $C \times C$ is just $2 - 2g$.  (This is a form of the Poincare--Hopf index theorem.)   The self-intersection of the graph of the hyperelliptic involution is also $2 - 2g$ (since it the image of the diagonal under the automorphism $(P,Q) \mapsto (P,\bar{Q})$ of $C^2$.
If we quotient out by the involution of $C^2$ given by transposition, to get 
$Sym^2 C$, then the self-intersection is halved, and so we get that the image 
of the graph of the hyperelliptic involution has self-intersection $1 - g$. 
