The riemann-surfaces tag has no usage guidance.

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### Properly Discontinuous Action

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...

**32**

votes

**3**answers

2k views

### If Spec Z is like a Riemann surface, what's the analogue of integration along a contour?

Rings of functions on a nonsingular algebraic curve (which, over $\mathbb{C}$, are holomorphic functions on a compact Riemann surface) and rings of integers in number fields are both examples of ...

**30**

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**5**answers

3k views

### Maryam Mirzakhani's works

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...

**30**

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**5**answers

2k views

### Intuition behind moduli space of curves

For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the ...

**20**

votes

**2**answers

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### Is there anything special about the Riemann surface $y^2 = x(x^{10}+11x^5-1)$?

I stumbled upon the fact that the Bolza surface can be obtained as the locus of the equation,
$y^2 = x^5-x$
Its automorphism group has the highest order for genus $2$, namely $48$. I recognized ...

**19**

votes

**6**answers

3k views

### Problem in Rick Miranda: finding genus of a Projective curve

I asked the following question in stack exchange (http://math.stackexchange.com/questions/21164/problem-in-rick-miranda-finding-genus-of-a-projective-curve) a few days ago, but didnt get any solution. ...

**19**

votes

**7**answers

3k views

### Links between Riemann surfaces and algebraic geometry

I'm taking introductory courses in both Riemann surfaces and algebraic geometry this term. I was surprised to hear that any compact Riemann surface is a projective variety. Apparently deeper links ...

**19**

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**10**answers

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### Uniformization theorem for Riemann surfaces

How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...

**17**

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**2**answers

940 views

### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...

**17**

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**0**answers

457 views

### On Determinants of Laplacians on Riemann Surfaces

History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...

**16**

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**9**answers

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### Teichmuller Theory introduction

What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?

**16**

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**5**answers

1k views

### Riemann surfaces: explicit algebraic equations

Suppose $\Gamma$ is a nice discrete subgroup of $SL(2,\mathbb{R})$ such that the genus of the Riemann surface $\mathbb{H}/\Gamma$ is larger than 1. We know that this Riemann surface is also an ...

**16**

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### “Physical” construction of nonconstant meromorphic functions on compact Riemann surfaces?

Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic ...

**16**

votes

**1**answer

349 views

### Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...

**16**

votes

**1**answer

762 views

### Irrational Numbers and the Riemann Surface of a Multi-Valued Function

Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not ...

**16**

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**1**answer

677 views

### Octonions and the Fano plane.

Does the Fano plane mnemonic for octonion multiplication have any deeper meaning?
http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg
The symmetry group of the Fano plane is PSL(2,7), ...

**14**

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**3**answers

2k views

### Is there an algebraic construction of the Quillen (determinant) Line Bundle?

Let's consider the moduli space of representations of $\pi=\pi_1(\Sigma)$ (a surface group) into $G$ (a lie group). Call this $X=\operatorname{Hom}(\pi,G)$, and let ...

**13**

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2k views

### Teichmuller modular forms and number theory

Do higher genus Teichmuller modular forms have, or are they expected to have, implications for number theory that generalize the sorts of results that flow from the study of classical modular forms?

**13**

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**4**answers

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### Representations of surface groups via holomorphic connections

EDIT: Tony Pantev has pointed out that the answer to this question will appear in forthcoming work of Bogomolov-Soloviev-Yotov. I look forward to reading it!
Background
Let $E \to X$ be a ...

**13**

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**2**answers

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### Automorphisms of Riemann Surfaces

Izumi Kuribayashi and Akikazu Kuribayashi have classified all groups of automorphisms of compact Riemann surfaces of genus 3,4,5. (J. Pure Applied Alg.65(3)-Sept.1990, and J. Alg.134(1) Oct.1990)
...

**13**

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**3**answers

863 views

### Injective morphism from curves to $\mathbb CP^2$

Is there a smooth compact complex curve that does not admit an injective holomorphic map to $\mathbb CP^2$ ? Let me stress, that the image of the curve in $\mathbb CP^2$ can have singularities.
I ...

**13**

votes

**1**answer

488 views

### Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...

**13**

votes

**2**answers

577 views

### Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...

**12**

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### What should be taught in a 1st course on Riemann Surfaces?

I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...

**12**

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**3**answers

564 views

### Hyperelliptic loci in Teichmueller spaces

Let ${\cal M}_g$ be the moduli space of smooth complex genus $g$ curves, let ${\cal H}_g\subset {\cal M}_g$ be the hyperelliptic locus and set ${{\cal H}}'_g$ to be the preimage of ${\cal H}_g$ in the ...

**12**

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**2**answers

880 views

### The fundamental group of a closed surface without classification of surfaces?

The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...

**11**

votes

**4**answers

629 views

### Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$?

I'm trying to motivate a bit of algebraic geometry in an abstract algebra course (while simultaneously trying to learn a bit of algebraic geometry), and I thought that it might be nice to present an ...

**11**

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**4**answers

413 views

### Growth of smallest closed geodesic in congruence subgroups?

Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.
How does the lower bound for the length of primitive geodesics on ...

**11**

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### A “holomorphic” Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.
I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...

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### Does every smoothly embedded surface $\mathbb{R}^3$ inherit a natural complex structure, and if so, which one?

Smoothly embed a genus g surface in $\mathbb{R}^3$, and pick a normal vector pointing "out" of the surface at each point. Then on each tangent plane, I have a map which rotates the tangent plane 90 ...

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624 views

### Conformal Welding Reference

I'm looking for a reference for the following fact: given two Riemann surfaces and an identification of their boundaries, once I topologically glue the surfaces together there exists a unique ...

**11**

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**1**answer

444 views

### Complex curves covered by smooth plane curves

Question: Is it true that for every smooth compact complex curve $C$ there exists a smooth curve $C'$ in $\mathbb CP^2$ that admits a non-trivial morphism (i.e. holomorphic map) $C'\to C$?
...

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### Canonical Immersion of the Double Torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...

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### Teichmuller theory and moduli of Riemann surfaces

This is a sequel to my earlier question asking for references for Teichmuller theory and moduli spaces of Riemann surfaces.
In this connection, I have read Chapter 11 of the book Primer of mapping ...

**10**

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732 views

### How do you recover the structure of the upper half plane from its description as a coset space?

This is maybe a dumb question. $SL_2(\mathbb{R})$ has a natural action on the upper half plane $\mathbb{H}$ which is transitive with stabilizer isomorphic to $SO_2(\mathbb{R})$. For this reason, ...

**9**

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**4**answers

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### Laplace-Beltrami Operator on Surfaces

I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature.
For instance,
What is the spectrum of the ...

**9**

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**3**answers

420 views

### Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$?
...

**9**

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**2**answers

395 views

### Embedding a Riemann surface in the sphere

Assume we have a Riemann surface, the underlying topological surface of which is a sphere with (possibly uncountably many) points removed. Can we always conformally embed this Riemann surface in the ...

**8**

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### What is a branched Riemann surface with cuts?

Edit: Let me restate the main claim being made in these two papers,
Consider the "branched" Riemann surface which has "n" sheets stuck along the intervals, $[z_i, z_{i+1}]$ for $i=1,..,2N$ then it ...

**8**

votes

**5**answers

727 views

### Schottky locus in genus 2

Let $\phi_g : \mathcal{M}_g \rightarrow \mathcal{A}_g$ be the period mapping from the open moduli space of genus $g$ Riemann surfaces to the moduli space of $g$-dimensional principally polarized ...

**8**

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### What prevents a cover to be Galois?

Let $f:X\rightarrow Y$ be a ramified cover of Riemann surfaces or algebraic curves over $\mathbb{C}$. My question is can one in terms of the ramification data of $f$, determine whether the cover is ...

**8**

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**1**answer

301 views

### Injective morphism from an elliptic curve to $\mathbb CP^2$.

Let $E$ be the elliptic curve $x^3+y^3+z^3=0$.
Question. Are there injective morphisms $E\to \mathbb CP^2$ of arbitrary high degree?
Comments. 1) There are injective morphisms $E\to \mathbb CP^2$ ...

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**4**answers

387 views

### Is there a non-abelian version of the Torelli map?

Let $C$ be a connected compact oriented real surface of genus $g$, let $G$ be a connected compact Lie group and let $G_\mathbb{C}$ be the complexification of $G$. One considers the moduli space $M ...

**8**

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**1**answer

349 views

### Given a curve, under which condition is the set of gonal morphisms finite

Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that ...

**8**

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**2**answers

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### Approximating holomorphic maps by holomorphic embeddings

Let $\mathrm{Hol}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the space of holomorphic maps of degree $d$ from a Riemann surface $\Sigma$ to complex projective space of dimension $n$. Let ...

**8**

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**2**answers

473 views

### Weil's theorem about maps from a discrete group to a Lie group.

Let K be a group (with discrete topology), G be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps ...

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**1**answer

592 views

### Is there an algebraic description of Yang-Mills measure?

Consider $X=\operatorname{Hom}(\pi,SL(2,\mathbb C))/\\!/SL(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,SU(2))/\\!/SU(2)$, where $\pi$ is a surface group. Note that if we use the right coordinates, ...

**8**

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**0**answers

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### Moduli space of semistable bundles

It is well-known that the space of S-equivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 surface M is $CP^3$ (more concretely ...

**7**

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**3**answers

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### Automorphisms of Riemann surface and mapping class

For a higher genus Riemann surface $\Sigma$, is it true that every nontrivial (holomorphic) automorphism is of nontrivial mapping class, i.e., not isotopic to the identity?

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### Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?
Edit: To rule out the case ...