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Locally looks like $z\mapsto z^n$, where $n$ is the branching index.

![](attachments/Pasted%20image%2020210612201758.png)

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Away from the branch locus, a map of Riemann surfaces is a covering map.

![](attachments/Pasted%20image%2020210612222033.png)

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Simple branching: $\mult_x(f) \leq 2$ for all $x$.

- Exercise: show that if $f: \PP^1(\CC)\to \PP^1(\CC)$ of degree $d$ has simple branching, then $f$ has $2d-2$ ramification points. Hint: apply Riemann-Hurwitz
-  Exercise: show that if $f:X\to Y$ is degree 2 where $g(X) = 1, g(Y) = 0$ then there are 4 branch points.
-  Exercise: consider $E:= \CC/\Lambda$ for $\Lambda = \ZZ + i\ZZ$.
	Quotient by $x\sim ix$ to get a map $f:E\to E/\sim$, and show $f$ has 
	- Two ramification points of multiplicity 4
	- Two ramification points of multiplicity 2
	What are their preimages in $\CC$?
	
- For $E = \CC / \Lambda$, show that for any $x\in E$ there is an $f\in \Aut_{\Hol}(E)$ with $f(x) = y$, so $\Aut_{\Hol}(E)$ is infinite.
- Cases of automorphisms of a lattice $\Lambda$ when $g(\CC/\Lambda) = 1$:
	- If $\Lambda \cong \ZZ + i\ZZ$ then $\Aut(\Lambda) = \ZZ/4 = \gens{z\mapsto iz}$.
	- If $\Lambda \cong \ZZ + \zeta_3 \ZZ$ then $\Aut(\Lambda) = \ZZ/6  = \gens{z\mapsto -\zeta_3 z}$
	- Otherwise $\Aut(\Lambda) = \ZZ/2 = \gens{z\mapsto -z}$. In this case $\Lambda \cong \ZZ + \zeta_m \ZZ$ for $\zeta_m$ some $m$th root of unity.
- Exercise: let $X = V(x^3 y + y^3z + z^3 x) \subseteq \PP^2(\CC)$, show that $g(X) = 3$.
	See [7.3 here](https://web.stanford.edu/~aaronlan/assets/riemann-surfaces.pdf)