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

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Canonical Embedding for Branched Cover

I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following: (i) $X$ has genus 4. (ii) The canonical embedding $\phi: X \rightarrow ...
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1answer
53 views

Proving the Existence of an Automorphism on $\mathbb{P}^{1}$

I recently came across the following problem while reading: Suppose that a compact Riemann surface $X$ has genus $g>1$. Let $\phi_{i}:X \rightarrow \mathbb{P}^{1}$ for $i=1,2$ be a pair of ...
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2answers
36 views

Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
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23 views

Diffeomorphism between a regular surface and the plane

Do Carmo states that (example 2, page 74) if $\mathbf x: U\subset\mathbb R^2\rightarrow S$ is a parameterization, then $\mathbf x^{-1}: \mathbf x(U)\rightarrow \mathbb R^2$ is differentiable. Why is ...
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1answer
41 views

The relation between principal curvature and curvature tensor?

To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I ...
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45 views

Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
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41 views

Parametrization of this elliptic curve

What's the simplest way to parametrize the curve given by the equation $$y^2 = (x^2-a^2)^2 - b^2,$$ namely simple functions (polynomials?) $x(z)$, $y(z)$, that would satisfy the above relation. This ...
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20 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
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Mittag-Leffler Problem

We have: $X$ a compact Riemann surface defined by $y^{2}=1-x^{6}$ and $P=(0,1) \in X$ a point given in local coordinates $(x,y)$. Furthermore, we have a meromorphic function $f(x,y)=y/x$ such that $f ...
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1answer
51 views

Existence of $g_{d}^{r}$ implies existence of which $g_{d'}^{r'}$'s?

Suppose I have a Riemann surface with a $g_{d}^{r}$. I am wondering what $g_{d'}^{r'}$'s exist for sure. For instance, since $h^{0}\left(L\left(-p \right)\right)$ is either $h^{0}\left(L\right)$ in ...
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55 views

Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem: If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most $3$ ...
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1answer
10 views

Branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$

Preamble: Probably my question will be highly downvoted and soon closed, because it is too simple. However I will make a tentative because I'm working on this for several days without finding any ...
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28 views

Constructing Riemann surfaces

At the risk of asking a question that has been already answered, I have been trying to figure out how to construct the Riemann surface of slightly more complicated examples, but after reading examples ...
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1answer
68 views

Weierstrass Point of a Riemann surface

I have that $X$ is a compact Riemann surface defined by the curve $y^{2}=1-x^{6}$ and a point $P=(0,1) \in X$ in the usual coordinates $(x,y)$. Ultimately, I want to solve a Mittag-Leffler problem on ...
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2answers
55 views

Forms on Riemann Surfaces

I want to show that the space of smooth $(1,0)$ forms on a compact Riemann surface $X$ has the natural splitting: $\mathcal{E}^{1,0}(X)=\Omega(X) \oplus \partial\mathcal{E}^{0}(X)$, where $\Omega(X)$ ...
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39 views

Why does the space of germs construction correspond to the gluing construction of Riemann surfaces?

I know this might be too broad / vague a question, but still looking for somebody to write something meaningful about this. When constructing the Riemann surfaces, why does the space of germs ...
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1answer
48 views

$d$-branched coverings and finite morphisms of degree $d$

Consider a smooth projective curve $X$ over $\mathbb C$ (so $X$ is a projective $\mathbb C$-scheme, integral, of finite type....), and let $t: X\longrightarrow\mathbb P^1_{\mathbb ...
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1answer
23 views

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 ...
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2answers
76 views

holomorphic function $f: \mathbb{C}\setminus [-1,1]\longrightarrow \mathbb{C}$ such that $f^2(z)=z^2-1$.

Prove there does not exist a holomorphic function $f: \mathbb{C}\setminus [-1,1]\longrightarrow \mathbb{C}$ such that $f^2(z)=z^2-1$ for all $z\in \mathbb{C}\setminus [-1,1]$. I really do not know ...
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1answer
65 views

Metric and Curvature on a Riemann Surface

We are given a smooth conformal metric $\rho=\rho(z)\left|dz\right|$ on a Riemann surface $X$. I have a few questions relating to this: (a) The local formula $R(\rho)=\Delta \mathrm{log}\rho dx\,dy$ ...
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Are two equivalent coverings of a Riemann surface also biholomorphic?

Consider a compact Riemann surface $X$. If $p_1:Y\longrightarrow X $ and $p_2:Z\longrightarrow X$ are two topological coverings of $X$, then it is univocally defined a structure of Riemann surface on ...
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1answer
29 views

Extending a biholomorphic map between two Riemann surfaces

Consider the following problem: $X$ and $Y$ are two compact Riemann surfaces, $S$ is a finite subset of $Y$ and $f:X\longrightarrow Y$ is a holomorphic map whose set of branch points is $S$. Now ...
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2answers
46 views

A more general definition of branched covering.

If $f:X\longrightarrow Y$ is a holomorphic map between two compact Riemann surfaces, then $f$ is called also a branched covering map. This because the branched points of $f$ form a finite set ...
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Applications of Belyi's theorem

Belyi's theorem (1979) is stated as follows: A smooth projective curve over $\mathbb C$ is defined over a number field if and only if there exists a finite morphism (of varieties over $\mathbb ...
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2answers
81 views

What are the things I need to know to study Topology

I am looking to learn about Riemann Surfaces but I know that beforehand I need to study certain subjects like Metric and Topological Spaces, Complex/Real Analysis and Complex Functions. Can anyone ...
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32 views

Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$ This is the same as giving the (very ample) line bundle over the curve ...
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62 views

The topology on the Riemann surface $\mathbb P^1(\mathbb C)$

In Miranda's book, is shown in detail that the projective line $\mathbb P^1(\mathbb C)$ is a Riemann surface. One considers simply $\mathbb P^1(\mathbb C)$ as a set (without any topology); then by ...
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2answers
44 views

Riemann Sphere/Surfaces Pre-Requisites

I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on ...
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1answer
20 views

Subgroups of PSL(2,R): criteria for discreteness

Let $G$ be a subgroup of $SL_2(\mathbb R)$ generated by a set of matrices $\mathcal M=(M_i)_{i\in I}$. Is there an effective criterion on $\mathcal M$ ensuring that $G$ is discrete?
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1answer
41 views

Biholomorphic maps from unit disc

Let $f$ be biholomorphic map from the unit disc onto some $D \subset \overline{\mathbb{C}}$ (considered as a Riemann sphere, so it is holomorphic) with $$f(z)=\frac{1}{z}+c_1z+c_2z^2+\cdots$$ What ...
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1answer
71 views

Cohomology to compute number of holes?

Can one use cohomology to compute the number of holes in a space $E$, where $E=R\times R$, $R$ is a Riemann surface of genus $g$, - i.e., is $\dim(H^n(E))$, and by Künneth's formula, $H^{n}(E) \cong ...
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1answer
90 views

Is there an algebraic invariant for complex curves that's mapped to injectively?

Consider the functor $\pi_1: \text{Closed Surfaces} \rightarrow \textbf{Grp}$. This is homotopy invariant; every closed topological surface has a unique fundamental group. In the reverse direction, by ...
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56 views

Cartesian product of Riemann Surfaces [closed]

I have a question regarding the Cartesian product of Riemann surfaces. Let $T^2$ denote the $2$ dimensional torus. $T^2=S^1\times S^1$. Also, let $T^4$ denote the $4$ dimensional torus. ...
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1answer
104 views

How to compute this Riemann surface?

This question is related to other more general question that I asked Computing Riemann surfaces of a given algebraic function. By the way, I've found an approaching in Markushevich's book that ...
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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 ...
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Rationality of divisors at infinity

In an attempt to clarify to myself some terminology (ant the scope of the Riemann-Roch theorem), I would like to ask for examples of genus $1$ curves of the form $$C : y^2 = ax^4 + b$$ where $a, b \in ...
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24 views

Pushforward of differentials (?) and Abel Jacobi map on principal divisors

Let $X$ be a complete one-dimensional curve over $\Bbb C$ (please add any hypotheses in case I forgot them). Then we have the Abel-Jacobi map $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
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20 views

Show that $d\log f$ is a 1-current on a 1-dimensional complex manifold

I am having trouble with this problem (and it might be because I have the formulation slightly off). I need to show that $d\log f$ is a 1-current on a 1-dimensional complex manifold $M$. This means ...
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62 views

Riemann-Roch theorem for singular curves

It might be a naive question, but I just realized I had not thought about this before. If $C$ is a smooth curve, for any line bundle $D$ we have the Riemann-Roch formula: $$\chi(D)=\deg D+1-g(C).$$ ...
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1answer
54 views

Definition of PU(2,1)?

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of ...
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1answer
23 views

Order of poles of y on y^2 = x^4 + a

The complex points on the curve $y^2 = x^4 + a$, together with two additional points $P^+, P^-$, can be viewed as compact Riemann surface $X$. What is the order of poles of the map $(x, y) \mapsto y$ ...
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13 views

Local coordinate of a (hyper)elliptic curve at infinity

I would like to ask for some help to clarify the following: In (9) of page 7 of http://page.math.tu-berlin.de/~bobenko/Lehre/Skripte/RS.pdf one finds stated that $(\mu, \lambda) \mapsto ...
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1answer
21 views

Rational/meromorphic functions on curves

Let, say, $E(\mathbb{C})$ be the set of affine points on an elliptic curve $y^2 = x^3 + ax + b$. Then $E(\mathbb{C})$, together with an additional point $\mathcal{O}$, can be viewed as a compact ...
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Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
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1answer
39 views

Why is an admissible function from a non-compact surface non-surjective?

I'm at the end of the proof of uniformization for simply connected manifolds in Farkas-Kra's Riemann Surfaces text. I feel like I'm missing something really obvious here. (The proof in question is on ...
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1answer
33 views

Homotopic mappings

Let $$ f_t(z)= \left\{ \begin{array}{ll} z & \mbox{if}\ \ 0<arg(z)< \pi/2-m_0 \\ z\ e^{-t/2m_0} & \mbox{if }\ \ \pi/2-m_0\leq arg(z) \leq \pi/2+m_0\\ z\ \exp(-t) ...
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1answer
45 views

Isomorphism between Riemann Surface and $\mathbb{P}^1$

Let $X$ be a compact Riemann surface. Prove that if $X$ is isomorphic to $\mathbb{P}^1$, then $X$ admits a meromorphic function $f$ that has a single pole and that this pole has multiplicity one. ...
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24 views

Why is every one dimensional Complex Manifold paracompact?

I read on the page Why are smooth manifolds defined to be paracompact? in one of the answers that every one dimensional complex manifold is automatically paracompact, i.e. there is no complex analogue ...
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1answer
69 views

Meromorphic functions a constant sheaf?

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic ...
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1answer
37 views

Extending maps on a Riemann surface

I came across the following definition for functions on a Riemann surface: A nonconstant analytic function on a Riemann surface, $f_{1}:X_{1} \rightarrow \mathbb{C}$ extends $f_{0}:X_{0} \rightarrow ...