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I am trying to solve the following but not getting anywhere.

Let $f:\Sigma_{1}\rightarrow \Sigma_{2}$ be a non-constant holomorphic map between compact Riemann surfaces $\Sigma_{i},i=1,2$.

(a) Prove that for any $p\in \Sigma_{2},f^{-1}(p)$ is a finite set and hence defines a positive divisor $D_p$ on $\Sigma_{1}$ of degree $deg(D_{p})=deg(f)$.

(b) If $\Sigma_{2}=\overline{\mathbb{C}}$ prove that the linear equivalence class of $D_{p}$ is independent of the choice of $p$.

(c) If $\Sigma_{2}\neq \overline{\mathbb{C}}$ is the linear equivalence of $D_{p}$ independent of the choice of $p$?.

I understand that (a) comes from the fact that the degree of such a map is finite and that the sum of multiplicities in the preimage must equal to the degree.

(b) I don't really know what to do. I would need to show for $p\neq p'\in \overline{\mathbb{C}}$ that there exists a meromorphic function $q$ on $\Sigma_{1}$ such that $D_{p}=D_{p'}+(q)$. Here $(q)$ is the divisor associated with the meromorphic function $q$. If $\Sigma_{1}$ was also the Riemann sphere then finding such a $q$ would be easy, one could just choose an appropiate function of the form $q(z)=(z-a_1)\dots (z-a_l)/((z-b_1)\dots (z-b_k))$...but for arbitrary $\Sigma_{1}$ it is a lot more difficult.

(c) I trust that knowing (b) would enlighten one as to how to approach (c). Please give at most a hint for this last part.

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Not sure this is the expected method but there is one

  • See your $f : \Sigma_1 \to \overline{\mathbb{C}}$ as a meromorphic function $f : \Sigma_1 \to \mathbb{C}$. Write $f^{-1}(0) =\{a_i\}_{i=1}^d$, $f^{-1}(\infty)=\{c_i\}_{i=1}^d$ and for a $p \in \mathbb{C}$, $\ f^{-1}(p)=\{b_i\}_{i=1}^d$ (the zeros of $f(z),f(z)-p,\frac{1}{f(z)}$ counted with multiplicity) so that $div(f(z)) = \sum_{i=1}^d a_i-c_i,div(f(z)-p) = \sum_{i=1}^d b_i-c_i$ and with $g(z) = \frac{f(z)}{f(z)-p}$ then $div(g(z)) = \sum_{i=1}^d a_i-b_i=D_0-D_p$

  • In other words $\overline{\mathbb{C}}$ works because for any two points $p,q \ne \infty$ there exists $\phi \in Aut(\overline{\mathbb{C}})$ such that $\phi(p) = q, \phi(\infty) = \phi(\infty)$ and there exists $z \mapsto \frac{1}{z}$ exchanging $0,\infty$.

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