# Questions tagged [riemann-surfaces]

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

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### How to compute the ramification indices in the degree-genus formula

I've been study the proof of the genus formula and I am a little confused about the ramification index of one of the maps involved. Let $C$ be a projective algebraic curve in $\mathbb{C}P^2$ such that ...
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### Special definition of “algebraic curve” for Riemann surfaces?

On the book Algebraic curves and Riemann Surface by Rick Miranda, page 169, I see the following definition: (Last part of definition 1.1) A complex Riemann surface $X$ is an algebraic curve if the ...
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### Is there a software with which you can visualise a complex function?

I just discovered that complex functions exist, and that you need four dimensions to represent them (i.e. three space dimensions and one represented by colours => Riemman’s surfaces). I was ...
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### The Monodromy Representation of a Riemann Riemann Surface with a Singular Projective Structure

Let $R$ be a Riemann Surface with a Regular Projective Structure $\nu$. Questions: What is the Monodromy Representation $\mu: \pi_1(R) \longrightarrow PSL(2,\mathbb{C})$, how to define it? If the ...
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### If E is non-singular then it has non-singular weierstrass equation

Let $E$ be a non-singular projective curve of genus one. There exist regular functions $x,y$ on $E$ satisfying a Weiestrass equation $$y^2 = x^3 + ax + b$$ Is this equation necessarily non-singular? ...
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### Equivalent Statements for the Fibre Bundle

Suppose $S$ is a Riemann Surface and $M$ is given as a $\mathbb{C} \mathbb{P} (1)-$bundle over $S$ where $\pi:M \longrightarrow S$ is the corresponding projection. Is not the previous statement (...
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### Question about how a divisor ensure zero in $l(D)$

Let $D$ be a divisor on a Riemann surface, prove if $d(D)<0$, then $l(D)=\{0\}$. Here $d(D)$ is the degree of $D$, $l(D):=\{f\quad meromorphic|(f)+D\geq0 \}$.
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### Unramified holomorphic map is isomorphism

In my course about Riemann surfaces, the professor briefly mentioned the following as a fact that we shall just accept: If $X$ is a connected, compact Riemann surface and $f:X\to\mathbb C_\infty$ is ...
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### Integration of complex $(p,q)$-form

In complex geometry, we have $(p,q)$form $\in$ $\wedge^{p,q}T^*X$, I wonder how to define their integration on submanifold, or top-form on all manifold. For instance, in the Riemann surface book I ...
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### Genus 1 Riemann surface admits meromorphic $f$ and $g$ satisfying $g^2=f^3+a_1f+a_2$

I am working on the following problem: Let $M$ be a genus 1 Riemann surface. By studying the spaces $L(-kp)$ for a point $p\in M$ and $k=1,\dots,6,$ show that $M$ admits meromorphic functions $f,g$ ...
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### Biholomorphism to Riemann sphere

Supose $X$ is a compact Riemann surface of genus 0 and suppose we have a meromorphic function $f:X\to \bar{\mathbb{C}}$. How can we see that $X$ is biholomorphic to the Riemann sphere?? I think it ...
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### Isomorphism of meromorphic function fields implies Riemann surfaces are isomorphic

This is about from Forster, Lectures on Riemann Surfaces, Exercise 8.1. The exercise is the following: ($\mathcal{M}(X)$ represents the field of meromorphic functions on $X$, and likewise for $Y$.) ...
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