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I have seen two ways to construct hyperelliptic curves, and it seems to me that the intuition behind the change of coordinate is not the same. I like better the second construction (which is pretty topological), but understand better the intuition of the first one (which is more analytical).

First construction. In my lecture notes, one constructs an hyperelliptic curve as follow : consider the Riemann surface $X \subset \mathbb C^2$ defined by $y^2 = p(x)$ for some polynomial $p(x) = (x-a_1) \dots (x-a_k)$ such that $a_i \neq a_j$ whenever $i \neq j$. Then, the holomorphic function $\pi \colon (x,y) \mapsto x$ is a ramified covering of degree 2 (the branching points being $a_1, \ldots, a_k$).

The idea is now to extend this ramified covering to a application $\tilde X \to \mathbb P^1$ of degree 2. In order to do that, observe that when $x \to \infty$, also $y \to \infty$, and let be two new local coordinates $z = 1/x$, $w = 1/y$. Then pulling back a small punctured disk centered in $0$ by $z$ and rewritting $y^2 = p(x)$ as $w^2 = 1/p(1/z)$, it appears that we can extend $\pi$ above $z = 0$ by one or two points depending of the parity of $k$. We then have the wanted $\tilde X$, compact Riemann surface.

Second construction. (Ref. Rick Miranda's Algebraic Curves and Riemann Surfaces) As before we have $X \subset \mathbb C^2$ and we also construct $Y \subset \mathbb C^2$ defined by $w^2 = z^{2m} p(1/z)$ where $k=2m$ or $2m-1$. Then let the biholomorphism $$\varphi \colon \{(x,y) \in X \mid x \neq 0\} \to \{(z,w) \in Y \mid z \neq 0\}, (x,y) \mapsto (z,w) = (1/x, y/x^m),$$ and we obtain $\tilde X$ by glueing $X$ and $Y$ over $\varphi$.

Question. What is the intuition behind the change of coordinates $z=1/x$, $w=y/x^m$ in the second construction ? It does not seems natural to me, whereas the change $z=1/x$, $w=1/y$ does. In other words, where does it come that we set $Y$ as the locus of $w^2 = z^{2m}p(1/z)$ ?

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