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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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What is the genus of $y=x^n$?

I would like to say that the Riemann surface S defined by $y=x^n$ has genus 0 but I don't know if I make a mistake anywhere, could you check if I'm right? First option: $Y: S \to \mathbb{P}^1$, $(x, y) \to y$ has degree n and has 2 branching points:…
Thomas
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How to use Riemann-Roch to solve this?

Let $C\in \mathbb{C}\mathbb{P}^{N}$ be a nonsingular complex curve of genus $g$. Let $p_{1},\dots,p_{k}$ be distinct points on $C$ and $n_{1},\dots,n_{k}$ positive integers. 1) Estimate an upper bound on the dimension of meromophic functions on $C$…
Bombyx mori
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when $G$ acts properly discontinuously on riemann surface then the orbits are closed sets?

could any one tell me how to prove: when $G$ acts properly discontinuously on riemann surface then the orbits are closed sets? $G$ is the group of all homeomorphism on $X$ say. so we have $f:G\times X\rightarrow X$ be a proper discont. action so…
balchhal
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non-equivalent Riemann surfaces of genus $1$

It is well known that all the compact orientable connected Hausdorff genus $1$ surfaces are homeomorphic, but they may have different complex structures. In fact, consider the following connected region in $\mathbb{C}$. $$G=\{ z\in\mathbb{C}\colon…
p Groups
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relation between order and multiplicity

Could any one tell me with an example, what is the relation between Order and Multiplicity of a holomorphic function on Riemann Surface,and how this formuale comes? for example let $p\in X$ be not a pole for $f$, and $f(p)=z_0$ then $f(z)-z_o$ has…
Myshkin
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$\mathbb{CP}^1$ is compact?

$\mathbb{CP}^1$ is the set of all one dimensional subspaces of $\mathbb{C}^2$, if $(z,w)\in \mathbb{C}^2$ be non zero , then its span is a point in $\mathbb{CP}^1$.let $U_0=\{[z:w]:z\neq 0\}$ and $U_1=\{[z:w]:w\neq 0\}$, $(z,w)\in \mathbb{C}^2$,and…
Myshkin
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genus of the quotient $g(X/G) \le g(X)$

Let $X$ be a Riemann Surface of genus $g(X)$ and $G$ a group acting holomorphically and effectively over $X$. I'm reading Miranda and he used twice the fact that the genus $g(X/G) \le g(X)$. He used this fact at least when $g(X)=0,1$. I don't know…
Eren
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tangent function determines a local homeomorphism

The question ask me to show that tangent function determines a local homeomorphism $\tan: \mathbb C \to \mathbb C P^1$. I don't understand what the question asking, is the question asking me to show that $\tan:\mathbb C \to \mathbb CP^1$ is a local…
SamC
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Meromorphic functions with two poles

It is well known that a Riemann surface $C$ such that there exists a meromorphic function with just one simple pole on it, then $C $ is the Riemann sphere. What can be said if there exists a meromorphic function with exactly two simple poles? And…
Carlo
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Comparing genus of domain and image for maps between Riemann surfaces.

I've been asked to show that if $R$ and $S$ are compact, connected Riemann surfaces, and $f: R \to S$ is holomorphic then $g(R) \ge g(S)$ (g is the genus). Now surely this fact follows from the fact that $f$ is an open map hence $f(R)$ is clopen…
Wooster
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Rank of canonical bundle of a Riemann surface

I am currently learning about Riemann surfaces, and just recently learned about line bundles. I've been asked to show on an assignment that $K_X=\cup_{x \in X} \Omega_x$ is a holomorphic line bundle on X, called the canonical line bundle on X. Here,…
Garnet
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Meromorphic functions on the Riemann Sphere - why can I assume we have no pole at $\infty$?

On the bottom of page 19 of Milne's Modular Forms notes (http://www.jmilne.org/math/CourseNotes/MF.pdf), he says Let $g$ be a meromorphic function on $S$. After possibly replacing $g(z)$ by $g(z-c)$ we can assume $g$ has neither a zero nor a pole…
Kumquats
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Field of meromorphic functions

Let $A, B$ and $C$ be compact Riemann surfaces with non-constant holomorphic maps $f: A \rightarrow C$ and $g: B \rightarrow C$, such that fields of meromorphic functions are isomorphic $M(A)=M(B)$ over $M(C)$. How to prove that $A$ bihilomorphic to…
evgeny
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Removable singularity theorems for manifolds

All the theorems on removable singularities are for functions defined on open domains $\Omega \in \mathbb{C}$. But what are the corresponding theorems for functions defined on Riemann surfaces? How do they differ and are there any extra issues we…
user82235
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Ramified covering of a compact Riemann surface

Let $C$ be a compact Riemann surface. Let $x_1,\ldots,x_n$ be $n$ distinct points on it. For $i=1,\ldots,n$, let $m_i$ be a positive integer considered as a `ramification order' at $x_i$. Questions: (1) are there conditions (and if yes, what are…
Tirlondon
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