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I am trying to understand the following example from the book Riemann surface by Donaldson (p.48). This was given after introducing the notion of monodromy of the covering.

Consider the Riemann surface $X$ defined by the equation $w^2=F(z)$, where $$f(z)=(z-z_1)(z-z_2) \dots (z-z_{2n}),$$ and where $F: X \to \mathbb C$ is the projection to the $z$ factor. Then $\Delta=\{z_1, \dots, z_{2n}\}$ and $\pi_1(\mathbb C \setminus \Delta)$ us generated by $2n$ loops $\gamma_1, \dots, \gamma_{2n}$, where $\gamma_i$ is a standard loop going once around $z_i$. The degree $d$ is $2$, and the representation $\rho$ maps each generator $\gamma_i$ to the non-trivial element of $S_2$ (a transposition of the two objects). In traditional language, we make cuts along $n$ disjoint paths joining $z_{2i-1}$ to $z_{2i}$ for $i=1, \dots, n$. Then we take two copies of the cut plane and form $X \setminus R$, $R=F^{-1}(\Delta)$ by gluing these along the cuts, More generally, we can express the procedure as saying that we make cuts so that $\rho$ becomes trivial on $\pi_1$ of the cut plane, and then $\rho$ is just the combinatorial data required to specify the gluing along the cuts.

Questions

  1. Why the representation $\rho$ maps each generator $\gamma_i$ to the non-trivial element of $S_2$? Since the degree of $F$ is $2$, we are considering $S_2$ but why non-trivial element?

  2. Why there are even number of zeros for $F$? What's wrong with odd number of zeros?

  3. I don't understand the concept of cut planes well. Why $X\setminus R$ can be obtained by gluing two copies of the cut plane?

  4. I don't understand "we make cut so that $\rho$ becomes trivial on $\pi_1$. What does it mean?

I appreciate any help. Thank you.

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    $\begingroup$ To start with simple (yet highly illustrative) cases, how well do you understand the Riemann surfaces $w^2 = z$ and $w^2 = 1 - z^2$? This MO thread may be helpful: mathoverflow.net/questions/135819/… $\endgroup$ – Andrew D. Hwang Aug 14 '13 at 15:30

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