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Questions tagged [riemann-surfaces]

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

Riemann surfaces are one-dimensional complex manifolds that are deformations of $\mathbb{C}.$

Riemann surfaces are orientable as a real manifold, and every simply connected Riemann surface is conformally equivalent to one of the following:

Elliptic (Positive curvature): The Riemann sphere (the complex plane with an extension to a point at infinity)

Parabolic (zero curvature): The complex plane

Hyperbolic (negative curvature): The open disk or upper half-plane

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Normalization of Abelian differentials of the second or third kind on Riemann surfaces seems not well defined

I am currently studying compact Riemann surfaces (more specifically hyperelliptic surface), and have a problem understanding the definition of the normalization of differentials of the second or third kind. This normalization seems dependent on the…
J.G.Q
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For all $d \geq 1$ there exist a torus $X= \mathbb{C} / \Lambda$ and a holomorphic map $X \rightarrow X$ of degree $d.$

Prove that for all $d \geq 1$ there exist a torus $X= \mathbb{C} / \Lambda$ and a holomorphic map $X = \mathbb{C} / \Lambda\rightarrow X= \mathbb{C} / \Lambda$ of degree $d.$ Attempt: Let $\Lambda$ denote the lattice of $\mathbb{C}$ generated by $1,…
user73681
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Application of Riemann-Roch in genus 3

Referring to the notation of Application of Riemann-Roch how is it possible to show that in genus $g=3$ then $X$ is a double covering space of the Riemann sphere ramified in 8 points or it is isomorphic to a curve in $\mathbb{P}^2(\mathbb{C})$ of…
balestrav
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Question on Forster's proof of the residue theorem

I have some questions about the proof of the residue theorem in Lectures on Riemann Surfaces by Otto Forster. The Residue Theorem. Suppose $X$ is a compact Riemann surface and $a_1,\cdots,a_n$ are distinct points in $X$. Let $X':= X \setminus…
Aki
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Order of poles of y on y^2 = x^4 + a

The complex points on the curve $y^2 = x^4 + a$, together with two additional points $P^+, P^-$, can be viewed as compact Riemann surface $X$. What is the order of poles of the map $(x, y) \mapsto y$ at the $P^+, P^-$? (it seems to equal 2 at both…
user132176
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Rational/meromorphic functions on curves

Let, say, $E(\mathbb{C})$ be the set of affine points on an elliptic curve $y^2 = x^3 + ax + b$. Then $E(\mathbb{C})$, together with an additional point $\mathcal{O}$, can be viewed as a compact Riemann surface $X$. Let one call a meromorphic…
Albertas
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Isomorphism between Riemann Surface and $\mathbb{P}^1$

Let $X$ be a compact Riemann surface. Prove that if $X$ is isomorphic to $\mathbb{P}^1$, then $X$ admits a meromorphic function $f$ that has a single pole and that this pole has multiplicity one. Since $X$ and $\mathbb{P}^1$ are isomorphic, then…
user44322
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Riemann Surface of $z^3$

I'm new to the topic of Riemann surfaces and complex analysis, and I tried to compute the Riemann surface of $z^3$ on Wolfram Alpha but couldn't because its not a multivalued function. How can I compute the riemann surface of $z^3$ or I got…
Krcx
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The monodromy representation of the projection map from a Fermat Curve

I have been trying to solve the following problem for quite some time now: Let X denote the Fermat curve of degree d in $\mathbb{P}^2$, defined by the homogenous polynomial $$x^d+y^d+z^d=0$$. Let $F:X \rightarrow \mathbb{P}^1$ be defined by…
Dedalus
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Riemann surface $w^3= z^2(z^2-1)$

How to find the chart around $(0,0)$ of $\{(z,w)\in\mathbb C\times \mathbb C: w^3= z^2(z^2-1)\}$? For $f(z,w)= w^3-z^2(z^2-1)$, in this case both $\frac{\partial f(z,w)}{\partial z}$ and $\frac{\partial f(z,w)}{\partial w}$ at $(0,0)$ vanishes. So…
zapkm
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Meromorphic function on a Riemann surface with associated holomorphic map

There is a lemma in Rick Miranda's Book "Algebraic Curves and Riemann surfaces " is the following, Let $f$ be a meromorphic function on a Riemann surface $X$ , with associated holomorphic map $F:X\rightarrow \mathbb{C}_{\infty} $ , show that , If…
Infinity
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Meromorphic function on Riemann surface

I've got this exercise I can't solve. May someone help me? Thank you. Let $X, Y, Z$ be homogeneous coordinates on complex projective plan and let $C=\{[X:Y:Z] |X^{4}+XY^{3}+Z^{4}=0\}$. Consider the meromorphic function $f=\frac{X}{Y}$ defined on…
Simone
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Analytic embedding of complex plane in Riemann surfaces with finite complement

$\mathbb{C}$ can be analytically embedded in the Riemann sphere such that the complement of the image of this embedding is only one point. Are there any other Riemann surfaces such that we can embed $\mathbb{C}$ in it such that the complement is…
Constant
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Can someon help me find missing figures in papers by Bujalance and Kuusalo?

There are two papers that i am trying very hard to understand. They concern the period matrices of hyperelliptic curves and are just easy enough so that i have chance of understanding them. They are as follows: "Period Matrices of…
teeshirt jones
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Why is this a Riemann surface?

I am reading chapter 3 of Donaldson's Riemann Surfaces. In the end of this chapter he tries to show that any discrete subgroup $\Gamma$ of Aut(D) can make D/$\Gamma$ into a Riemann surface. I understand that every point in D/$\Gamma$ has a…
Zichen Gao
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