# Symmetric product of genus 2-surface

Let $\Sigma$ be the genus 2-surface.

Denote $\operatorname{Sym}^2(\Sigma)=\Sigma\times \Sigma/\mathbb{Z}/2$, where $\mathbb{Z}/2$ acts on $\Sigma\times \Sigma$ by $(x,y)\mapsto (y,x)$.

In the very famous paper of Ozsvath and Szabo (see page 1035), the authors said that $\operatorname{Sym}^2(\Sigma)$ is the blow up of $T^4=S^1\times S^1\times S^1\times S^1$ at one point. I want to prove this in detail. (Detail is omitted in the Original paper.)

We have an Abel-Jacobi map, $\mu\colon \operatorname{Sym}^2(\Sigma)\to \operatorname{Pic}^2(\Sigma)$, given by $(p,q)\mapsto [p+q]$.

On the other hand, from the Euler sequence, $$0\to H^1(\Sigma;\mathbb{Z})\to H^1(\Sigma;\mathcal{O})\to H^1(\Sigma,\mathcal{O}^*)=\operatorname{Pic}(\Sigma)\to H^2(\Sigma;\mathbb{Z})=\mathbb{Z}\to 0$$

we can identify $\operatorname{Pic}(\Sigma)=\mathbb{Z}\times H^1(\Sigma;\mathcal{O})/H^1(\Sigma;\mathbb{Z})$ and by looking at $\operatorname{Pic}^2$ part only, we have $\operatorname{Pic}^2(\Sigma)=H^1(\Sigma;\mathcal{O})/H^1(\Sigma;\mathbb{Z})=H^1(\Sigma;\mathbb{R})/H^1(\Sigma;\mathbb{Z})=T^4$.

Hence, Abel-Jacobi map, define above can be regarded as $\mu\colon \operatorname{Sym}^2(\Sigma)\to T^4$. Ozsvath-Szabo claim that $\mu$ is the blow up and exceptional sphere $S$ is given by $$S=\{[x,\tau(x)] ~|~ x\in \Sigma,\tau\colon \Sigma\to \Sigma \textrm{ is the hyper-elliptic involution}\}.$$

However, I could not prove $\mu$ is a blow up.

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