# Questions tagged [riemann-surfaces]

For questions about Riemann surfaces, that is complex manifolds of (complex) dimension 1, and related topics.

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### Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
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### $\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation. I read that this equation had a $\tau$-structure,...
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### How to write holomorphic line bundles on compact Riemann surfaces in coordinates

I would like to write down coordinate charts for holomorphic vector bundles on a smooth complex (compact) curve $C$. For example, if $C = \mathbb C \mathbb P^1 = \{ [x_0 : x_1 ] \}$, let \begin{...
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### Given a curve over $\mathbb{P}^1$ how can I determine the monodromy around a ramification point?

Given a smooth curve of the form $$f(y) - g(x)$$ over $\mathbb{A}^1_x$ with a smooth compactification $C \to \mathbb{P}^1$, how can I determine the monodromy of the points of the fibers? Are there ...
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### Where can I find precise examples of ramified coverings of $\mathbb{C}P^1$?

One of the definitions of simple Hurwitz number $h_{g,\mu}$ is that it counts up to automorphisms the number of ramified coverings of $\mathbb{C}P^1$ such that covering space is a connected surface of ...
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### Relationships between the field of meromorphic functions and the fundamental group

Let $X$ be a connected compact Riemannian surface. Then the ring of meromorphic functions on $X$ is a field and we denote it as $M(X)$. Let $E$ be a finite extension over $M(X)$. Then we can show that ...
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### $\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ as interior of compact Riemann Surface with Boundary

A takehome exam problem for my Riemann Surfaces class, which used Griffith's Introduction to Algebraic Curves, was the following: Show that $S=\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ is not interior ...
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### Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
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### Polynomials for Bicubic Planar Dessin d'Enfants

A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in ... a plane. For the coloring to exist, the graph must be bipartite. ... The ... embedding may be ...
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### How to tell if a simply-connected curve is the complex plane or disk

Suppose I have an analytic function $f: \mathbb{C}^2 \to \mathbb{C}$ whose zero locus $V=\{(z,w) \in \mathbb{C}^2 : f(z,w)=0\}$ is smooth and simply connected. By uniformization, $V$ is conformally ...
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### How to show that the Weierstrass function is meromorphic in whole of $\mathbb{C}$

The Weierstrass function is defined as: $$\wp(z) = \frac{1}{z^2}+ {\sum\limits_{\omega\in\Gamma'}} \left (\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}\right).$$ Fix $r>0$. For $\left|z\right| \leq r,$...
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### Riemann surfaces and folding

I am learning about Riemann surfaces as covering, and was interested to know how Riemann first thought of them. Looking at his collected papers, I found the following definition in his lecture called “...
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### Understanding Complex Differentials (forms)

In the study of Riemann surfaces, many books bring in their discussions, the complex differentials or differential forms, and there my understanding gets stopped. I personally interacted with many ...
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### Obtaining a single-valued branch of $\ln \left( \frac{z-a}{z-b} \right)$ with a branch cut

It is rather easy to see that the function $$f(z) = \ln \left( \frac{z-a}{z-b} \right)$$ has branch points at $z=a$ and $z=b$, My question is why considering a branch cut "connecting" $a$ and $b$ ...
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### Not taking holomorphic bundles for granted

When we say something to the effect of, "Consider a holomorphic bundle $V$ on a complex manifold $X$...", we are saying that the transition functions of $V$ are holomorphic functions with respect to ...
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### Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
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### Number of fixed points of automorphism on Riemann Surface

I want to prove that the number of fixed points of a non-identity automorphism on a compact Riemann surface $X$ is at most 2g+2. Following hints given, I have considered the divisor $D = (g+1)P$, ...
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### Reference about Moduli Spaces of Riemann surfaces

I'm looking for some (introductory, and in any case not too technical) reference (book, lecture notes, papers) regarding moduli spaces $\mathcal{M}_g$ and $\mathcal{M}_{g,n}$ of (punctured) Riemann ...
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### Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
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### Spinors on algebraic plane curves

I'm interested in parameterizing spinors on Riemann surfaces. For my purposes, it's best to represent the Riemann surfaces as immersed in $\mathbb{C}P^2$, i.e. as algebraic plane curves. Apparently, ...
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### How do we define a complete metric on a Riemann surface with punctures?

This question is related to another question. If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this ...