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To my understanding, we can consider a hyper-elliptic curve $y^{2}=f(x)$ (with $f$ a degree $2n$ polynomial) as a curve branched over $2n$ distinct branch points chosen on the Riemann sphere $\mathbb{P}^{1}$. The branch cuts will connect these branch points pairwise. Now, I saw posted on an informal blog entry that the moduli space of these hyper-elliptic curves is given by,

$$\rm{Conf}_{2n}(\mathbb{P}^{1}) / \rm{Aut}(\mathbb{P}^{1})$$

where $\rm{Conf}_{2n}(\mathbb{P}^{1})$ is the configuration space of $2n$ distinct points on $\mathbb{P}^{1}$. This seems to be telling me, if correct, that the moduli space of these curves doesn't depend at all on the shape of the branch cuts which connect pairwise the branch points.

Is this true? It seems like it fails to account for cases where the branch cuts behave poorly like intersecting with each other, or something.

Also, am I correct in saying that $\rm{Aut}(\mathbb{P}^{1}) = \rm{SL}(2, \mathbb{C})$? Thanks!

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