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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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How do I see Riemann sphere is a Riemann surface

How do one show that Riemann sphere $=\mathbb{C} \cup\{\infty\}$ has a unique structure of Riemann surface? To show $\mathbb{C} \cup\{\infty\}$ is a Riemann surface I just need to set up two charts, they are $\{U_1,f_1\}$ and $\{U_2,f_2\}$…
SamC
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Riemann Surface of Genus 1

Let $X$ be a compact Riemann Surface of genus 1. Let $Cl_0(X) := \frac{Div(X)}{PDiv(X)}$, where $PDiv(X)$ is the subgroup of principal divisors on $X$. Let $P \in X$ be a fixed point. We have a bijection $i_p : X \rightarrow Cl_0(X)$ given by $X \in…
random123
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Riemann surface for $w = f(z) = \sqrt{z^2}$

To obtain Riemann surface for $w = f(z) = \sqrt{z}$ we get two copies of $z$-planes with cuts. After they are joined $f(z)$ gives us a one-to-one correspondence between this Riemann surface and $w$-complex plane. Now, Riemann surface for $w = f(z) =…
user75619
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Correspondence between bitangents of a quartic and odd theta characteristics

Let $C$ be a Riemann surface of genus $g=3$. I can't understand why the following statements are true: If $C$ is not hyperelliptic, then the canonical series $|K|$ embeds $C$ as a smooth quartic in $\mathbb{P}^2$ The odd theta characteristics of…
Davide
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Finding the Riemann surface of $w = z^{1/2}$

I'm trying to find the Euler characteristic of $R = \{(z,w) : f(w,z) = w^2 - z = 0\}$. To do this I'm using the Riemann-Hurwitz theorem with the projection $\Pi: R \to \mathbb{C}P^1$. Now in local coordinates we can write this map as $\pi (w) =…
Wooster
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Question on Fulton's coverage of Riemann surfaces

Riemann surfaces beginners question: (I am learning about normalization of algebraic curves for the first time using Fulton's Algebraic topology and was doing fine until i hit a this snag) SHORT FORM OF QUESTION: Would somebody who has a copy…
teeshirt jones
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Finding degree and branching index of projection map

I'm looking at the Riemann surface $R = \{(z,w) : f(z,w) = w^3 - z^3 + z = 0 \}$. I'm looking at the projection map $f: (z,w) \to z$ and I'm trying to find the degree and the branching index. I can see that whenever $w \ne 0$ we have local…
Wooster
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Investigate the covering of the sphereby the sphere associated with the rational function

Here there is an example of Singerman Complex Functions, i think i understand it, the point is that having $f$ degree 3, do i have three copies of $\mathbb{C}$? if so, how can i glue them? What i want to know is how to draw the surface $S$ such that…
USER
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Creating a holomorphic map from a three holed torus to a two holed torus.

I'm trying to make a non-constant holomorphic map, $f$, between a 3 holed torus and a 2 holed torus, with no branch points. Now I can see that $deg(f) = 2$ from the Riemann Hurwitz formula. So intuitively, I want to treat the three holed torus as…
Wooster
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Equivalence of atlases on Riemann surface

I've got a very vague question about Riemann surfaces, more like a meta-question: One of the first things one defines is, obviously, an atlas on a Riemann surface $R$ ($R$ is a connected Hausdorff topological space), i.e. a family of open sets…
Fedra
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Holomorphic differential on a compact Riemann surface

Let $X$ be a compact surface of genus $g\ge 2$. I want to show that there exists $P_1,\dots,P_g$ distinct points on $X$, such that a holomorphic differential $\omega\in H^0(X,\Omega_X^1)$ must be zero if $\omega(P_j)=0,j=1,2,\dots,g$. I know a fact:…
save123
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How does the problem of classification of Riemann surfaces extend beyond the uniformization theorem?

If the uniformization theorem gives a 'first classification' of Riemann surfaces into hyperbolic, parabolic, and elliptic surfaces, then what would a 'second classification' look like? If it makes any sense to consider such a thing... I know that…
George
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A confusion in the proof of the dimension of $q$-differentials $\mathscr{H}^q(M)$ and the result in torus

I get that the dimension for $\mathscr{H}^q(M)$ is $(2q-1)(g-1)$ for $q > 1$ by the Riemann-Roch theorem. Here $M$ is a compact connected Riemann surface with genus $g > 0$ and $\mathscr{H}^q(M)$ is the vector space consisting all the holomorphic…
onRiv
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Is $\sqrt[3]{z^2}$ a Riemann surface including $(0, 0)$?

I'm wondering if $\sqrt[3]{z^2}$ is a Riemann surface including point $(0, 0)$ or not. Treat it as the zero set of $F(z, w) = 0$, where $F(z, w) = z^2-w^3$. Then $F_z = 2z$ and $F_w = 3w^2$ and both are zero in point $(0, 0)$. From this view point…
onRiv
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the universal covering space of compact Riemann surface(how the covering map be holormporphic?)

I occured in this question when I was learning the proof of the Riemann-Roch theorem for effective divisors with the vedio here(Sorry for this vedio is in Chinese, though the blackboard is written in English). Suppose we have a compact Riemann…
onRiv
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