TITLE: Second symmetric product of a hyperelliptic curve
QUESTION [3 upvotes]: Let $C$ be a hyperelliptic curve of genus $g\geq 3$, let $C^{(2)}$ be the second symmetric product of $C$ with itself, i.e. the quotient of $C\times C$ by the involution $(p,q)\mapsto (q,p)$ and let $g_2^1$ be the unique degree two linear series on $C$. The map
$$
 \mu\colon C^{(2)}\to JC \qquad p+q\mapsto [p+q-g_2^1]
$$
contracts the rational curve in $C^{(2)}$ defined by the pairs $p+\sigma(p)$, where $\sigma\in {\rm Aut}(C)$ is the hyperelliptic involution. The surface $S =\mu(C^{(2)})$ is invariant under the change of sign map $x\mapsto -x$ of $JC$. My question is what is known about the quotient surface
$$
 K = S/\langle -1\rangle.
$$
Is there any reference to this in literature?
I am omitting the case $g=2$ since in that case a lot is known being $K$ the Kummer surface of $C$.

REPLY [1 votes]: Your surface $K$ is singular, so maybe you want to consider its natural desingularization $\tilde{K} $ which is the quotient of $C^{(2)}$ by the involution induced by $\sigma $. The double covering $C\rightarrow \mathbb{P}^1$ induces a map $C^{(2)}\rightarrow (\mathbb{P}^1)^{(2)}\cong \mathbb{P^2}$ which factors through a double covering $\pi :\tilde{K}\rightarrow \mathbb{P}^2 $. The ramification points are the points $w+p$ with $w$ a Weierstrass point of $C$; it follows that the branch locus  of $\pi $ consists of $2g+2$ lines, which are all tangent to the conic $\Delta$ in $\mathbb{P}^2$ corresponding to the diagonal.  So $\tilde{K} $ is the double cover of $\mathbb{P}^2$ branched along these $2g+2$ lines; and $K$ is obtained from $\tilde{K} $ by blowing down the pull back $\pi ^{-1}(\Delta )$, which is isomorphic to $C$.
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