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

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1answer
34 views

Show that something is a subsystem of a complete linear system

I have a simple and basic question concerning degree of projective curves and I'm referring to something I've read on Miranda's book, Algebraic curves and Riemann Surfaces, Chapter VII, 3. The Degree ...
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1answer
9 views

Sum over the branches of a composition of an entire function with the branches of an algebraic function is entire.

Let $l(t)$ be the solution of the polynomial equation $g(l,t)=\det(lE-(A-tB))=0$, where E is the identity and A, and B are $n\times n$ matrices. The natural domain of definition is a Riemann surface ...
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1answer
49 views

Divisor of meromorphic section of point bundle over a Riemann surface

Let $X$ be a compact connected Riemann surface (not $\mathbb{P}^1$), $p\in X$ be a point on it. Let $L$ be the holomorphic line bundle associated to the divisor $D=p$. By construction $L$ comes with a ...
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0answers
38 views

fundamental group and covering space

I have a question about the fondamental group of the following covering space $$ p : Y \rightarrow X ; \; Y \owns (x,y) \mapsto x \in X $$ where $X = {\mathbb P}^1$ and $$ Y := \{(x,y) \in {\mathbb ...
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60 views
+100

Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let ...
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0answers
45 views

Proof of Riemann-Roch using Mittag-Leffler

In the introduction for Rick Miranda's book "algebraic curves and Riemann surfaces", it says that they will prove Riemann-Roch "in an algebraic manner, via an adaptation of the adelic proof, expressed ...
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0answers
27 views

Why is this function on the Riemann surface holomorphic?

Forster defines analytic continuation of a germ of a holomorphic function at a point on a Riemann surface as follows. Suppose $ X$ is a Riemann surface, $a\in X$ and $\phi\in\mathcal{O}_a$ is a ...
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15 views

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 ...
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2answers
74 views

How to define the complex square root $ \sqrt{z} $?

We need to define the complex square root $ \sqrt{z} $ on a small open $ U \subset \mathbb{C} $, for example a disc. Let put : $ \mathcal{F} (U) = \{\ f: U \to \mathbb {C} \ / \ f \ \text{is ...
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0answers
20 views

Creating a holomorphic map from a three holed torus to a two holed torus.

I'm trying to make a non-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 ...
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1answer
37 views

Discrete faithful representation in $PSL(2,\mathbb R)$ and horocycles in hyperbolic space

Let $S$ be a closed oriented surface of genus $g>1$. Is the following true ? Let $\alpha,\beta\in \pi_1(S)\backslash \{1\}$ and $\rho:\pi_1(S)\rightarrow PSL(2,\mathbb R)$ be a discrete ...
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2answers
38 views

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 ...
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2answers
72 views

Maps between Riemann surfaces are open and continuous

I'm having some trouble with a couple of concepts in Riemman surfaces that I would really appreciate some help clarifying! Firstly, is it true that a holomorphic map between two Riemann surfaces $f:R ...
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2answers
63 views

computation on hyper surface $z=x^2+y^2$

I have problem with following exercise Consider the hypersurface $M$ parametrized by $z=x^2+y^2$. Endow this with the Riemannian metric induced from the $\mathbb{R}^3$. Compute the sectional ...
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1answer
35 views

Intuitively understanding Riemann surfaces

I'm looking at the Riemann surface of $f(z) = z^{1/2}$ so the set $\{(z,w) \in \mathbb{C}^2 : w^2 = z \}$. I understand that the point of the riemann surface is to understand this multi-valued ...
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0answers
14 views

Is it possible to give a metric to a simply connected Riemann surface?

I would like to know if, given a simply connected Riemann surface $\mathcal{S},$ is possible to define a metric on it.
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0answers
15 views

How does one define the universal curve over the moduli space of stable curves

In relation to the Prym class. How does one define the universal curve $\pi:\bar{\mathcal{C}_g}\longrightarrow\bar{\mathcal{M}_g}$ for genus $g\geq 2$ and their dualizing sheaf $\omega_g$.
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1answer
108 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
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0answers
16 views

Does there exist a complex analytic map between compact Riemann surfaces with certain conditions?

I am wondering if there exists a complex analytic map $\pi:X\rightarrow Y$ with $g(Y)=g(X)-1.$ By Riemann-Hurwitz formula, I think the simplest of such maps is a complex analytic map between compact ...
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0answers
59 views

Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
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1answer
64 views

Riemann surface from $x^2 + y^2 = 1$ for $x,y \in \mathbb{C}$

I am reading Edward Frenkel's book Love and Math. In Chapter 9, it is talked about the one-to-one correspondence of solution of algebraic function of complex numbers and Riemann surfaces. can anyone ...
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1answer
38 views

Conformal classes and almost-complex structures

It is well-known that on closed oriented surfaces $S$, conformal classes of metrics on $S$ correspond bijectively to complex structures on $S$. My understanding is that this correspondance goes as ...
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0answers
49 views

How can a finite graph be viewed as a discrete analogue of a Riemann surface?

In the paper "Riemann–Roch and Abel–Jacobi theory on a finite graph" by Baker and Norine, the first line of the abstract states: "It is well known that a finite graph can be viewed, in many respects, ...
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1answer
47 views

(Path)Connectedness of $w^{2} = \sin z$.

I recently began to study Riemann surfaces, and I got a problem while checking some examples in the book. It is easy to see, for example, that the subset $\{(z, w)\in \mathbb{C}^2\mid w^{2} = \sin ...
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3answers
59 views

what does endowed mean?

A surface becomes a Riemann surface when it is endowed with an additional geometric structure. English is not my native language, and I cant understand what does endowed mean here.
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1answer
26 views

Finding poles of quadratic differentials

Consider the following quadratic differential (on a Riemann surface): $$ \phi_1\left(z\right)=\frac{P_4\left(z\right)}{\left(z-1\right)^2\left(z-a\right)^2}\frac{\mathrm{d}z^2}{z^2} $$ Here, ...
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2answers
46 views

Cover of a finitely punctured plane

Let $X_n$ be the plane with a finite number $n$ of punctures, and let $p : Y \rightarrow X_n$ be a covering map (it may have infinite degree). Can we say anything about the topology of $Y$? (I know ...
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2answers
50 views

Multiplicity of an holomorphic map between Riemann surfaces

I need help understanding the meaning of multiplicity in a point of an holomorphic map between Riemann sufaces. So $F\colon X \to Y$ be an holomorphic, not constant map between Riemann surfaces and ...
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1answer
65 views

Reading Griffiths Harris: Quick question

Why is a meromorphic section without zeros and poles on a compact Riemann surface necessarily a constant? Thank you very much.
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0answers
35 views

What is the circumference and area of a circle with radius r?

Let $M$ a Riemann surface with constant Gauss curvature $K = K_{0}$. Calculate the circumference and area of a circle with radius $r$. Also, calculate the geodesic curvature $K_{g}((r)$ of a circle ...
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0answers
59 views

Branched coverings of unit disk

Is there a classification of branched coverings of the closed unit disk $\mathbb{D} =\{z\in \mathbb{C} \ | \ |z| \leq 1 \}$? Here we consider only branched covering projections which restrict to ...
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1answer
51 views

Using the Riemann Hurwitz Formula

I am working with the function $f(z)=\frac{z^3}{1-z^2}$ from the Riemann Sphere to itself. I'm trying to show that this satisfies the Riemann-Hurwitz formula given ...
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1answer
55 views

maple plot of Belyi function

I would like to understand how to construct Figure 5 of the paper Composition is a generalized symmetry by Alexander Zvonkin: The hypermap/dessin d'enfant of Figure 4 is while the Belyi function ...
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0answers
16 views

Pre-requisites for 'The Concept of a Riemann Surface'

I just got my copy of Hermann Weyl's 'The Concept of a Riemann Surface'. On the first page, the author uses the principle of analytic continuation that tells me I should read Complex Analysis before ...
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23 views

Injectivity of a map from a homotopy set to a homology group

Let $\Sigma$ be a closed Riemann surface and $X$ a 1-connected topological space. I would like to prove the following fact. (A) The map $[\Sigma,X] \to H_2(X;\mathbb Z)$ defined by $[f] \mapsto ...
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1answer
34 views

non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$ F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\} $$ and call it non-singular if ...
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29 views

Two unit disks spliced together?

$$ x^2 + (y-z)^2 = 2 x^2 z/y $$ The surface represented by above equation is formed by radial cuts on two separate unit diameter disks spliced together forming a "continuous" surface around ...
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1answer
24 views

Pullback and quadratic differentials on Riemann surfaces

Suppose we have some meromorphic quadratic differential $q=\phi\left(x\right)\mathrm{d}x^2$ on a punctured Riemann surface $\mathcal{C}$, and that $q$ is the pullback via a function $f$ of a ...
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12 views

constant $K$ and $k_g$ ovals growth

Referring to my recent post: Ovals of constant $ k_g$ on constant $K$ surfaces, using geodesic polar coordinates with radial geodesic lines built along v=constant around a fixed point on a constant ...
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1answer
27 views

What is the Weil-Petersson metric of the moduli space of elliptic curves?

One can define the Weil-Petersson metric on the moduli space of Riemann surfaces. I would like to know an explicit example of such a metric. What is the Weil-Petersson metric of the moduli space of ...
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40 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
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0answers
35 views

Compact manifold with smooth structure

Is the following surface smooth and compact, when all its partial derivatives are continuous? How to tell about self-intersections without visualization? $x=\cos(a) + \cos(a + b) + \cos(a + b + ...
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1answer
57 views

A question from Otto Forster's book

I'm tackling exercise 8.2 on page 59 which goes as follows: Let $X$ and $Y$ be compact Riemann surfaces, $a_1,\dots,a_n\in X$ and $b_1,\dots,b_m\in Y$ distinct points and ...
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41 views

Holomorphic and meromorphic functions on Riemann surfaces

On any domain $\Omega\subset \mathbb{C}$, the set of all holomorphic functions form an integral domain. Its field of quotient is the set of all meromorphic functions on $\Omega$. However this is not ...
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44 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
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0answers
35 views

Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational ...
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1answer
39 views

Holomorphic map or Riemann suface into projective space, Miranda's book

I have the following question after reading Chapter V, prop. 4.3 of Miranda's book Algebraic Curves and Riemann Surfaces. The setting is as follows: we have a Riemann surface $X$ and a holomorphic ...
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3answers
376 views

What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ...
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0answers
16 views

Finding an algebraic equation given divisors

I'm trying to find an algebraic curve that represents a specific Riemann surface and my question goes like this: Given divisors $(\omega_1) = P_1 + 5 P_2 + 2 P_3,$ $(\omega_2) = 5 P_1 + P_2 + 2 P_3,$ ...
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2answers
78 views

Group of automorphisms of a compact hyperbolic Riemann surface is finite

Let $M$ be a compact hyperbolic Riemann surface. Is there a simple way to show that the automorphism group $Aut(M)$ of conformal self-mappings of $M$ is a finite group? Recall that a hyperbolic ...