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Questions tagged [riemann-surfaces]

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

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Quotient Space of Hausdorff space

Is it true that quotient space of a Hausdorff space is necessarily Hausdorff? In the book "Algebraic Curves and Riemann Surfaces", by Miranda, the author writes: "$\mathbb{}P^2$ can be viewed as the ...
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2answers
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How do different definitions of “degree” coincide?

I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do ...
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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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Why the emphasis on Projective Space in Algebraic Geometry?

I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what little ...
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1answer
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Toward “integrals of rational functions along an algebraic curve”

In a talk by V.I. Arnold, this is said: When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-...
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$\mathbb{C}\mathbb{P}^1$ is homeomorphic to $S^2$

I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In ...
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2answers
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On the Usual Orientation of Cubic Graphs in Random Construction of Riemann Surfaces

In "Random Construction of Riemann Surfaces", Robert Brooks and Eran Makover say : Definition 2.1 A left-hand turn path on $(\Gamma, \mathcal O)$ is a closed path on [the cubic graph] $\Gamma$ such ...
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What is the best way to see that the dimension of the moduli space of curves of genus $g>1$ is $3g-3$?

This fact was apparently known to Riemann. How did Riemann think about this?
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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 $\mathbb C$...
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Canonical divisor on algebraic curve

Can someone help me with this problem? Let $D$ be a divisor on an algebraic curve $X$ of genus $g$, such that $\deg D = 2g-2$ and $\dim L(D) = g$. Then $D$ must be a canonical divisor. By Riemann-...
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Space of Germs of Holomorphic Function

A bit of a general question, but here goes. Morally, what is the space of germs of a holomorphic function? I know that a germ is simply an equivalence class of function elements, where we regard two ...
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Books on complex analysis

Is there any book on $1$-dimensional complex analysis, where all is written in the language of sheaf theory? It's clear, that a lot of constructions can be formulated in simplier way using it. There ...
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Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

Let $\pi:C^{'} \rightarrow C$ an unramified double cover of a complex Riemann surface $C$ of genus $g$. With the symbol $Nm_{\pi}$ we mean the norm application that takes a meromorphic function on $C^{...
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1answer
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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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Operational details (Implementation) of Kneser's method of fractional iteration of function $\exp(x)$?

For a long time (a couple of years) I'm following the Q&A's about "half-iterate of $\exp(x)$" etc. where there exists a $\mathbb C \to \mathbb C$ due to Schröder's method, but also a $\mathbb R \...
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Hecke operators acting on the Jacobian $J(X_1(N))$

I'm stuck with p.238 of Diamond's book on modular forms. If $\{f_j\}_{j=1}^g$ is the eigenform basis of $S_2(\Gamma_0(N))$, fixing $p_0 \in X_0(N)$ we have an holomorphic map $$\phi : X_0(N) \to J(X_0(...
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Extending isomorphism of punctured Riemann surfaces

In Schlag's book on Riemann surfaces, we have Problem $4.15$, which is: Let $M, N$ be compact Riemann surfaces and suppose $f: M\setminus\mathcal{S} \to N\setminus\mathcal{S}'$ is an isomorphism, ...
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Hyperbolic metric geodesically complete

Consider the upper half plane model of the hyperbolic space ($\mathbb{H}$ with the riemannian metric $g=\frac{dx^2+dy^2}{y^2}$). It is known that $(\mathbb{H},g)$ is geodesically complete, which means ...
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Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface

This question is sort of an extension to this previous question of mine, Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve If one knows the multiplicity of a ...
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1answer
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Show that $\int_{\partial P}z\frac {f'(z)} {f(z)} dz $ is on the lattice $\Lambda$

Problem: Let $f(z)$ be a meromorphic function on the complex torus $\mathbb C/\Lambda$ that as a function on $\mathbb C$ has no zeros and no poles on $\partial P$, the boundary of the fundamental ...
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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 is because the branched points of $f$ form a finite set $S\...
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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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Is this complex structure unique?

Let $G$ be a subgroup of $PSL(2,\mathbb{C})$, so that $G$ acts on $\mathbb{C}\cup\{\infty\}$ by linear fractional transformations. We say that $G$ acts properly discontinuously at a point $z\in \...
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Show that a function from a Riemann Surface $g:Y\to\mathbb{C}$ is holomorphic iff its composition with a proper holomorphic map is holomorphic.

I'm trying to show the following: Let $f:X\to Y$ be a proper holomorphic map between connected, non-empty Riemann Surfaces. Show that a map $g:Y\to\mathbb{C}$ is holomorphic if and only if its ...
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Perspectives on Riemann Surfaces

So, I have come to a somewhat impasse concerning my class selection for next term, and I have exhausted all the 'biased' sources. So, I was wondering if anyone in this fantastic mathematical community ...
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Understanding Ramification Points

I really don't understand how to calculate ramification points for a general map between Riemann Surfaces. If anyone has a good explanation of this, would they be prepared to share it? Disclaimer: I'd ...
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1answer
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The importance of modular forms

I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called $E(\Lambda)=\...
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1answer
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Describing Riemann Surfaces

We've just learned about Riemann surfaces in my complex analysis class and to get a better understanding, I've been trying to find similar problems. I came across this: Describe the Riemann surfaces ...
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Complex analysis book with a view toward Riemann surfaces?

I am considering complex analysis as my next area of study. There are already a few threads asking about complex analysis texts (see Complex Analysis Book and What is a good complex analysis textbook?...
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What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?

Let $X$ be a compact Riemann surface of genus $g>1$, $f\in Aut(X)$, a biholomorphism of $X$ onto itself, $x\in X$ a fixed point of $f$. Since tangent map of a holomorphic map (on the real tangent ...
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Constant Curvature Metric and Biholomorphic Equivalence

This is probably a dumb question, but let's try it anyway. I know two versions of the uniformization theorem, and I don't understand their equivalence. The first says that every Riemann surface has a ...
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Good book for Riemann Surfaces

I am considering reading one of 'Algebraic curves and Riemann Surfaces' by Rick Miranda or 'Lectures on Riemann Surfaces' by Otto Forster. Which one of these is more advanced and comprehensive ? What ...
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2answers
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Can there be a point on a Riemann surface such that every rational function is ramified at this point?

Let $X$ be a compact connected Riemann surface, and let $S\subset X$ be a finite subset. Does there exist a morphism $f:X\to \mathbf{P}^1(\mathbf{C})$ which is unramified at the points of $S$? I'm ...
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Computing the monodromy for a cover of the Riemann sphere (and Puiseux expansions)

Given an irreducible polynomial $P(X,Y) \in \mathbb{C}[X,Y]$, one obtains by analytic continuation a Riemann surface $M$ with a branched covering $f \colon M \to \mathbb{P}^1_{/\mathbb{C}}$, ...
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Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves

I have two highly-coupled questions concerning holomorphic line bundles, and so I will go ahead and ask them together. The first concerns line bundles on $\mathbb{CP}^1$ and the other concerns line ...
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orientability of riemann surface

Could any one tell me about the orientability of riemann surfaces? well, Holomorphic maps between two open sets of complex plane preserves orientation of the plane,I mean conformal property of ...
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Riemann surface with punctures corresponds to a hyperbolic surface with cusps

I am reading a paper on Riemann surfaces and the author used the fact that $\{$Riemann surfaces with genus $g$ and $n$ punctures$\}$ is in one-to-one correspondence with $\{$ hyperbolic surfaces ...
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1answer
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Can a meromorphic function be written as ratio of holomorphic function?

Well, I want to know whether a meromorphic function can be written as ratio of two holomorphic function on $\mathbb{C}$ or on a Riemann surface. Thank you for help.
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Projective closure of an algebraic curve as a compactification of Riemann surface

Assume $f \in \mathbb{C}[x,y]$ a polynomial such that the affine algebraic curve $X=V(f)$ has no singular points. Then there is a natural structure of non-compact Riemann surface on $X$, which can be ...
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1answer
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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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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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1answer
810 views

Application of Riemann-Roch

The Riemann-Roch theorem states $l(D)-l(K-D)=1 + deg(D) - g$ where $D$ is a divisor on a compact Riemann surface $X$ and $K$ is the divisor of meromorphic 1-form. In the case $g=2$ and $D=K$ this is $...
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Connected sums and their homology

Edit: I already received a good answer to my second question. I'd be interested in a hint about the first one, as well. Thanks in advance! I'm interested in compact Riemann surfaces and their ...
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Why is a holomorphic map between compact connected Riemann surfaces a branched covering?

I have seen it claimed that a non-constant holomorphic map $f:X \rightarrow Y$ between compact connected Riemann surfaces is a branched covering i.e. surjective and there is a finite set $\Sigma \...
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Can $\mathbb RP^2$ cover $\mathbb S^2$?

I know $\mathbb{S}^2$ is the universal cover of $\mathbb{R}P^2$, but can $\mathbb{R}P^2$ be a covering space (at all) of $\mathbb{S}^2$? Attempt at solution It's clear that for a umramified covering ...
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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
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Euler characteristic of a connected sum of surfaces.

How do I see that the Euler characteristic of a connected sum of surfaces $S_1$ and $S_2$ is given by$$\chi(S_1 \# S_2) = \chi(S_1) + \chi(S_2) - 2?$$
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Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
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1answer
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Motivation for Mobius Transformation

Let $S$ denote the Riemann Sphere. Recall that a Mobius transformation is a function $f:S \to S$ defines as $z \to \frac {az+b}{cz+d}$ where $a,b,c,d \in \mathbb C$ with $ad-bc=1$. What is the ...
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Hairy ball theorem : a counter example ?

Recall the hairy ball theorem : any continuous vector field on a sphere $S^n$ of even dimension must vanish at least once. Now consider the Riemann sphere $\mathbb{C} \cup \infty \simeq S^2$. Let $X(...