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

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1answer
36 views

Attempt at understanding Weierstrass points

I'm reading through Springer - Riemann surfaces and Farkas and Kra - Riemann surfaces and theta functions. I'm attempting to get an understanding of Weierstrass points. I've come up with a (hopefully) ...
2
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0answers
61 views

Understanding the “shape” of a singular Riemann surface

Consider the singular Riemann surface given by the following expression: $$z^d w^d-z^d-w^d+t=0\ ,$$ where $t$ is a parameter in $(-1,1)$ and $d$ is a positive integer greater than 2. For $t\neq0$ the ...
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2answers
21 views

Quotient of space and a group of maps, Riemann surfaces

I've been attempting to study Riemann surfaces, and I have continuously run into this notion which eludes me. I see people write things like $ \mathbb H / <z\mapsto z+1>$ or $\mathbb D / PSL$. I ...
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0answers
19 views

Lemma in Farkas/Kra: Riemann surfaces on construction of domain satisfying certain properties

I'm having some trouble understanding the proof of this lemma. I can follow the construction of $u$ and $D$, however the final step in the proof seems to be without justification. Specifically why ...
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0answers
20 views

Corollary of Wolpert lemma

Recall Wolpert Lemma. Let $S$ be a surface with genus greater than 2, let $[X,f]$ and $[Y,g]$ two points of $T(S)$ (Teichmüller space) and let $\phi \colon X \to Y$ a $K$ quasi conformal homeo. Then ...
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0answers
15 views

Determining if an equation represents (?) a Riemann surface

This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of ...
6
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1answer
72 views

Animation of Weierstrass $\wp$-function as a map from a torus to the sphere?

I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can ...
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0answers
15 views

Is the structure constant additive on connected components?

Definition of the Structure Constant Let $M$ be a Riemann surface and $\mu$ a smooth metric on it; let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and ...
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0answers
59 views

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 ...
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1answer
52 views

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 ...
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1answer
22 views

Extending functions on proper subsets of $\mathbb C$ to functions on proper subsets of $S^2$.

There are a number of nice results about extending holomorphic and meromorphic functions from the complex plane $\mathbb C$ to the Riemann sphere $S^2$. See for instance Does entire function extend ...
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0answers
32 views

Base of homology on a Riemann surface and holomorphic differentials

I have two questions: 1) Given a Riemann surface $X$ of genus $g$ and an holomorphic differential $\omega$ on $X$, is it always possible to find a base $\{\delta_i\}_{i=1,\dots 2g}$ of ...
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1answer
16 views

Confusion about definition of ramification points in Forster

In the book "Riemann surfaces" by Forster a ramification point of $f:Y\to X$ is defined as a point $y\in Y$ such that there is no neighborhood $V$ of $y$ such that $f|_V$ is injective. On the other ...
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0answers
120 views

$\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation. I read that this equation had a ...
0
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1answer
38 views

Riemann surfaces from a number theoretic point of view

I need to learn the basic theory of Riemann surfaces and would like to pick a book which is most relevant to algebraic number theory. I have a good understanding of all underdraduate algebra and the ...
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0answers
10 views

Finiteness of valuation function defined in Levelt filtration.

I'm studying the Levelt filtration and a certain valuation function comes up and I'm trying to understand when (and why) it is finite. Let $S$ be a disk in the complex plane centred at $0$, $S' = S ...
2
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1answer
35 views

Proof of Hopf's theorem using Liouville

Hopf Theorem A topological sphere immersed as a constant mean curvature surface in $\mathbb{E}^3$ is a round sphere In Heinz Hopf's Differential Geometry in the Large, a proof is given of the ...
1
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1answer
44 views

Why the flat torus cannot be immersed in euclidean plane?

I am trying to prove the following claim: The flat $2$-dimensional torus cannot be isometrically immersed into $\mathbb{R}^2$ with the standard metric. That is, there is no immersion $f:T^2 ...
2
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1answer
32 views

Question about two definitions of Teichmuller space for a surface of genus $g$

There are many equivalent definitions of Teichmuller space for a surface of genus $g\ge 2$. One of them concerns the complex structure: the Teichmuller space $\mathcal{T}(g)$ is the set of the ...
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0answers
8 views

Convex standard fundamental polygon

The section of the wikipedia page "Fundamental Polygon" claims that one can construct a convex standard fundamental polygon from a metric fundamental polygon. I do not know a construction which ...
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0answers
20 views

Equivalence of Definitions of Quasiconformal Surfaces?

I have been reading John H. Hubbard's book $\textit{Teichmüller Theory vol. 1}$ and I am a little bit concerned with his definition of Quasiconformal Surface. Definition: A Quasiconformal surface ...
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1answer
89 views

The Weierstrass-Enneper representation, the Gauss map

Lemma: Let $x:S\to\mathbb{R}^3$ be a conformal minimal immersion of a Riemann surface. The 1-forms $f_k=(x_{k,u}-ix_{k,v})dz$ satisfy: $$ \sum_kf_k^2=0\qquad (1)\qquad \&\qquad ...
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1answer
27 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
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1answer
26 views

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 ...
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0answers
24 views

Inverse of a constant function on an open set

I was working on holomorphic functions and Riemann surfaces, and I was wondering about the inverse of a constant function: Let $f:U\rightarrow V$ be a holomorphic function between two Riemann ...
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0answers
75 views

2 Definitions of Holomorphic functions on Riemann surfaces

In a lecture that I currently attend we defined Riemann surfaces and holomorphic mappings on it somewhat different than in another lecture that I attended a year ago. My question is: Are these ...
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0answers
22 views

Equation of the curve corresponding to a principal polarization

Let $\mathbb{C}^2/\Lambda$ be a principally polarized abelian surface. I think it is well-known how to write down the equation of the divisor (Riemann surface) corresponding to the polarization, in ...
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0answers
21 views

Proving Riemann-Hurwitz formula for riemann sphere

Given a rational map $f:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$, where $\hat{\mathbb{C}}$ is the Riemann sphere, I need to prove that $2\deg(f) - 2 = \sum (v_f(p)-1)$, i.e. prove the Riemann-Hurwitz ...
4
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1answer
59 views

Link between Riemann surfaces and Galois theory

In my notes for a Geometry of Surfaces course that I'm studying, there is the following quote: (For those of you who like algebra and Galois theory) Studying compact connected Riemann surfaces is ...
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0answers
15 views

relation between degree and residues

Let $C$ a compact riemann surface of positive genus and $\omega_C$ the canonical divisor over $C$ with standard degree $2g-2$. Take on $C$ a divisor of positive degree $d$ and set ...
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2answers
95 views

$\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$ possible?

Is it possible to have $\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$? My question comes from the link beetween covering and field extensions. For covering the simplest example is ...
2
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1answer
66 views

Finding holomorphic map on Riemann surface from a map between two Riemann surfaces

I have a non-constant degree two map between Riemann surfaces $R$ and $S$, $f: R \to S$. I'm trying to find a holomorphic homeomorphism $\tau: R \to R$ such that $f(\tau) = f$ and $\tau^2$ is the ...
2
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1answer
67 views

Sheaf, étalé space with Riemann surfaces.

Let $f:X\rightarrow Y$ be an holomorphic map betwen two Riemann surfaces and let: $\Gamma:=${ $(x,y)\in X\times Y|y=f(x)$ } $\subset X\times Y$ be the graph of $f$. I have to show that ...
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1answer
76 views

Riemann Roch Meromorphic section on a line bundle.

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha w)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
2
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1answer
39 views

How to prove that the flat torus is indeed flat?

The $n$-dimensional torus can be obtained as a quotient: $T^n=\mathbb{R}^n/\mathbb{Z}^n$. As pointed out here, the standard metric on $\mathbb{R}^n$ is invariant under translation by the elements of ...
2
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1answer
80 views

Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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0answers
24 views

Automorphisms of simple covers of Riemann surfaces

Can anybody give me a simple proof that simple covers of a Riemann surface have no covering automorphisms?
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0answers
20 views

Linearity of map between holomorphic tangent spaces

The proposition I am trying to prove is as follows: Let $f:X\to Y$ be a holomorphic map between two Riemann surfaces. For each $x\in U$ the map $$ Df(x):T_{X,x}^{1,0}\ni [h]_x\mapsto [f\circ ...
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0answers
16 views

Local conformal coordinates on a surface

Let $\mathcal{M}\subset\mathbb{R}^3$ be a smooth enough regular surface. We want to show that around a point $p\in\mathcal{M}$ there is a neighborhood about $p$ in $\mathcal{M}$ which is parametrized ...
3
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1answer
83 views

Divisor on curve of genus $2$

I suffer from lack of concrete examples in Algebraic Geometry, so I will appreciate it if somebody can help me in understanding a bit better this one: Let $\mathcal{C}$ be a genus $2$ curve ...
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0answers
48 views

Mapping a curve into projective space

Let $\mathcal{C}$ be a (smooth, complex, projective) genus 2 curve. Take two different points $p,q\in\mathcal{C}$ and let $K$ be the canonical divisor class. I know (by means of Riemann-Roch) that the ...
2
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1answer
43 views

$j$-invariants of isogenous elliptic curves

Suppose that $E,E'$ are isogenous smooth complex elliptic curves - is there some relation between their $j$-invariants?
3
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1answer
61 views

Difference between Euler characteristics of a Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g$. Let $U$ be the complement of $r$ points in $X$. The Euler characteristic of $X$ = $2-2g$. That I understand. But I'm confused about the ...
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0answers
15 views

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 ...
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1answer
31 views

Surface constructed using curves

Suppose that $E$ and $F$ are two complex compact Riemann surfaces with genus greater or equal than $2$. Set $$S=E \times F$$ the surface composed by the cartesian product of thees curves. What can i ...
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12 views

Torus-like Riemann surface for a genus 1 Cassini oval

Among the Cassini ovals there is the lemniscate of Bernoulli. This latter curve has genus $0$ and can be mapped to a standard Riemann sphere: $$ (R(t+t^3), R(t-t^3), (1+t^4)/\sqrt{2}) \qquad t\in ...
0
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1answer
42 views

Distance on riemann sphere [duplicate]

Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} ...
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0answers
30 views

a question about finding umbilical points in an elipsoid.

Determine the umbilical points of the elipsoid $${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.$$ My thoughts: let $x=asin(\theta)cos(\phi),y=bsin(\theta)cos(\phi),$and $z=cos(\theta)$. Thus, I ...
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1answer
50 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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37 views

Rational section of the canonical line bundle of a smooth curve

Let $C$ a complex Riemann surface with genus $g>0$, $L$ a theta characteristic on $C$ i.e $L \in Pic(C)$ such that $L^2 \equiv \omega_C$ where $\omega_C$ is the caninical line bundle on C and ...