Questions tagged [riemann-surfaces]

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

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Positive divisor and Isomorphism to $\mathbb{C}_{\infty}$

I am trying to prove that if $X$ is compact a Riemann surface with a positive divisor $D$ such that $dim L(D)=1+deg(D)$, then $X\cong \mathbb{C}_{\infty}$. First we need to see that there exists $p\in ...
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Pull back of the $\mathcal{O}(1)$ bundle under the canonical map

Usually, this is used in many places, so I just want to understand it properly. For a non-hyperelliptic Riemann surface $S,$ we have the canonical embedding $\iota_K:S\rightarrow \mathbb{P}^{g-1}.$ ...
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The residues of a meromorphic form cancel, the defects of a Fuchsian ODE cancel - Coincidence?

Two facts that seem similar: The sum of the residues of a meromorphic differential $f(z) dz$ on a compact Riemann surface is zero. Fuchs relation: The sum of all indicial roots at all singularity ...
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Showing by Hand that $H^1(0)=0 , H^1(-p)=0$ in $\mathbb{C}$ , Mittag-Leffer Problems in $\mathbb{C}$ and $\mathbb{C}/L$

I am trying to show by hand, without using the Riemann-Roch Theorem, that in $\Bbb C_{\infty}$ we will have that $H^1(0)=0$ and $H^1(-p)=0$, but I am having some trouble constructing the functions ...
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Functions that separate points and tangents on an Algebraic Curve

I am trying to do an exercise from Rick Miranda's Book that goes like this Let $X$ be an algebraic Curve, show using the compactness of $X$ that there are a finite number of global meromorphic ...
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17 views

Holomorphic functions on Riemann surfaces with boundary

Suppose that $\Sigma$ is a compact Riemann surface with boundary and that $f: \Sigma \rightarrow \mathbb{C}$ is holomorphic*. If $f$ is real-valued along $\partial \Sigma$, is it necessarily true that ...
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Explicit construction of Riemann surface

Are there some book or online lecture/notes which explicitly constructs the (compact) Riemann surface associated to some polynomials/holomorphic functions ? I know the algorithm to do so, but I don't ...
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42 views

Transcendence degree of $\mathcal{M}(x)$ in $\mathbb{C}$, and $\mathcal{M}(X)$ is finitely generated over $\mathbb{C}$

I have been reading about Riemann surfaces from Rick Miranda's Book and now the term transcendent degree has come in to play, but I have no knowledge of field theory so I dont quite understand it I ...
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61 views

(geometric) intuition behind divisor class group

I'm taking classes in algebraic geometry and riemann surfaces, and in both these classes the divisor class group of a (certain kind of) scheme/riemann surface $X$ is defined roughly by "divisors mod ...
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Exercise 2.2 on Griffiths' [an introduction to algebraic curves]

I have no idea with the exercise 2.2 of Griffiths' nice book [introduction to algebraic curves], chapter 4, section 2, page 130. The problem is: if $j^*\omega=\omega$, then we can find some $\phi\in \...
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44 views

Show that every curve of genus 2 can be expressed as a fourth degree plane curve possessing a double point.

Show that every curve of genus 2 can be expressed as a fourth degree plane curve possessing a double point. This curve is of course a hyperelliptic curve. In order to find a map into $\mathbb{P}^3$, ...
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Fixed points of automorphism of unit disk

I'm reading the book 'A Course in Complex Analysis and Riemann Surfaces' by Wilhelm Schlag but I'm stuck at the following statement in section 4.8: Groups of Möbius transformations. We have that an ...
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31 views

Laurent series Approximation in Algebraic Curves

I am reading Rick's Miranda book and he's now talking about how we can do a laurent series approximation in an Algebraic curve,page $173$, that is Suppose that $X$ is an algebraic curve, fix a ...
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10 views

Defining a holomorphic map via a Linear System

I have been reading Rick's Miranda book on Riemann surfaces and now he states Let $Q \subset |D|$ be a base point free linear system of (projective ) dimension $n$ on a compact Riemann surface $X$. ...
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23 views

relation between Riemann theta function and Jacobi theta function

So we know Jacobi 3rd theta function can be defined using different summations such as: \begin{equation} \theta_{3}(a,b)=1+2\sum_{m=1}^{\infty}b^{m^2}\cos(2ma) \end{equation} and I also know that ...
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How to construct a degree $4$ polynomial $h(z)$ such that $h(z)$ has a triple root and $h(z) - 1$ has a double root?

This is a homework question, so please do not give me the full answer. I only need a hint that pushes me in the right direction. I have been asked to construct a holomorphic function $h(z)$ whose ...
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60 views

If $X$ is closed then $JX$ is closed

Let $(S, g, J)$ be a closed Riemann surface with a Riemannian metric $g$ compatible with the complex structure $J$. Suppose that a smooth vector field $X$ on $S$ is closed, i.e., the $1$-form $\omega =...
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57 views

When might Fenchel-Nielsen twist coordinates exceed 1/4?

When a compact Riemann surface of genus $g$ is cut up along $3g-3$ disjoint geodesic loops into $2g-2$ pairs of pants, the result is often described by giving Fenchel-Nielsen coordinates: one length ...
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33 views

Existence of a Green's function equivalent to existence of bounded harmonic function

I want to show the following are equivalent on a Riemann surface $W$: Green's function $g_W (p, q)$ exists. There exists a bounded harmonic function on $W$. There exists a positive harmonic function ...
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Geodesic- the derivation of thei scalar product

I would like to know how follows the $1$st $=$ in the very last formula in the snippet below (i.e. one on the l.h.s. below) $$\frac{d}{dt}\langle\frac{dx}{dt},\frac{dx}{dt}\rangle=2\langle\frac{dx}{dt}...
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Finite Number of Global Meromorphic Functions Separating Points and Tangents on a Compact Riemann Surface

In Miranda's "Algebraic Curves and Riemann Surfaces," there is a problem that asks the reader the show, using the compactness of a Riemann Surface $X$, that there are only a finite number of global ...
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38 views

Evenly generated Fuchsian groups

Consider the upper half plane $\mathbb{H}$ and a Fuchsian group $\Gamma$ (i. e. a discrete subgroup of $PSL(2,\mathbb{R})$ that acts freely and properly discontinuously on $\mathbb{H}$. I want to ...
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25 views

Complex structures with trivial automorphism groups

Fix a compact real surface $S$ of genus $g \geqslant 2$ and look at the space $\mathcal{T}$ of complex structures on $S$ (up to isotopy). I know it has a manifold structure (I think it is called the ...
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Holomorphic line projection map in projective space. [duplicate]

Suppose we have a line $L$ in the projective space $ \mathbb{CP}^2$ and we choose a point $R \in \mathbb{CP}^2$, such that $R \notin L$. For any other point $P \in \mathbb{CP}^2$, we define $L_{PR}$ ...
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27 views

Showing something is not a Riemann surface

So suppose we have $X=\{(x,y)\in \mathbb{C} : y^2=x^2+x^3\}$ and i am asked to see that this is not a Riemann surface. Well we know that it is not going to be a smoth affine plane curve because at the ...
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70 views

Characterization of Fuchsian groups containing hyperbolic elements

I want to find the Fuchsian groups that acts on the upper half plane $\mathbb{H}$ to give $n$-holed torus $\mathbb{T_n}$. I am following the book Fuschian Groups by Svetlana Katok. There's this ...
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Group action of a Fuchsian group on the upper half plane

An $n$-holed torus $\mathbb{T_n}$ is quotient of the upper half plane $\mathbb{H}$ by properly discontinuous action of a subgroup of automorphism group of $\mathbb{H}$. The automorphism group of $\...
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Very Ample Implies Separation of Points and Tangents: Miranda

As a preface, this is not homework. I'm a professor, after all. I'm reading through Miranda's "Algebraic Curves and Riemann Surfaces," and getting caught up on one point from Chapter VI. First, ...
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One-Sided Inverse of a Complex Polynomial

I encountered the following question in complex analysis: Let $P(z)$ be a polynomial such that if $P(z)\in \mathbb{D}$ then $z\in \mathbb{D}$ and $P'(z)\neq 0$. Prove that there exists an analytic ...
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Riemann Surface of $z=\sqrt{w}$

Today I started learning about Riemann surfaces. In Gamelin's Complex Analysis, Gamelin states that the Riemann surface of $z=\sqrt{w}$ is "essentially a sphere with two punctures corresponding to $0$ ...
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Plucker's Formula proof

I just read a proof of Plucker's Formula in Rick's Miranda book that says A smooth projective plane curve of degree $d$ has genus $\frac{(d-1)(d-2)}{2}$. To do this he defines map $\pi : X \...
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42 views

Definition of holomorphic differential forms

Reading Forster's Lectures on Riemann surfaces I find his definition of holomorphic differential forms unpleasant. Let $X$ be a Riemann surface. He says that they are forms $\omega$ with value in $T^{...
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Equality for Dimensions of Vector Spaces of Meromorphic Functions

I am reading a proof in algebraic geometry where the following equality is used: $\text{dim} \: L(C,K_C) = \text{dim} \: L(C,K_C + P),$ where $L(C,D)$ is the vector space of meromorphic functions ...
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43 views

What are the best books to study riemann surfaces from

I have taken a course in algebra , analysis.Which book should I start with at the very beginning(which would be easy enough to tackle) and what are the other prerequisites I need to know?
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Conventions regarding multivalued function and Riemann sheets.

Good day all. Is the value of a composed multivalued function $F(g(z),h(z),k(z),..)$ on different Riemann sheets given only by its formula or, in addition to the formula one needs to provide the ...
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25 views

Meromorphic functions on the Riemann sphere , generalization of a result

I had an exercise that goes like this that i was able to solve but was wondering if it was true for more Riemann surfaces , hopefully something not isomorphic to $\mathbb{C}_{\infty}$. Let $f$ and $...
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Holomorphic form on Hyperilliptic Riemann surface

I am trying to check that if we have an Hyperilliptic Rieamnn surface $X$ defined by $y^2=h(x)$, then $\frac{dx}{y}$ is a holomorphic $1$-form on $X$ if $g\geq 1$. I think i was able to do this for ...
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48 views

Fiber of covering map is orbit of deck transformation group

Some definitions: Let $\lambda: \mathbb H \to \mathbb C \setminus \{-1, +1 \}$ be the covering map, i.e. $\lambda$ is surjective and for every $z \neq \pm 1$ there exists a neighborhood $V_z \subseteq ...
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Infinite number of Unramified Holomorphic maps on the complex Torus

I am trying to an exercise that goes like this: Show that there are an infinite number of Unramified Holomorphic maps $F:\mathbb{C}/L \rightarrow\mathbb{C}/L $ that are not automorphisms. Well we ...
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$\phi$ is a map defined on $C$, a compact Riemann surface of genus $g$ and $D$ a divisor with $d=\mathrm{deg}D \ge 2g+1$.

Suppose $C$ is a compact Riemann surface of genus $g$, $D \in \mathrm{Div}(C)$, and $$d = \mathrm{deg}D \ge 2g+1$$ Suppose $\{f_0,\dots , f_{d-g}\}$ is a basis of $\mathscr{L}(D)=\{f \vert (f) + D \...
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28 views

Why the map from complex torus to the projective algebraic curve is continuous?

I am following the proof to show that the complex torus is the same as the projective algebraic curve. First we consider the complex torus minus a point, punctured torus, and show there is a ...
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87 views

Remembering Riemann-Roch

Embarrassingly, I've always struggled to remember the form of the Riemann-Roch theorem for curves. Does anyone have any intuition to share about how to remember the some of the terms in the formula? ...
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34 views

Theta functions identites

I have been reading Rick Miranda's Book on Riemann surfaces and to indroduce meromorphic functions on the complex torues $\mathbb{C}$\ $L$ he talks about theta functions. I was able to see that $\...
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40 views

Hyperosculating point count

This is exercise IV.4.6(b) of Harshorne. Let $X$ be a (smooth) curve of genus $g$ embedded as a curve of degree $d$ inside $P^n$, $n\geq 3$, and not contained in any hyperplane. Then I would like to ...
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63 views

Finding points with valency $\nu_f(p)>1$ for a map between Riemann surfaces.

I am trying to do an exercise for a course about Riemann surfaces and for that, I need to find the valency of all points with respect to a function in other words, the branch points with their ...
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1answer
19 views

Intersection divisor of a homogeneous polynomial with a line in the projective plane

I am trying to do an exercise from Rick's Miranda Book that goes like this Show that if $X$ is a line in the projective plane, then the intersection divisor of any other line with $X$ has degree ...
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1answer
22 views

Meromorphic function is holomorphic on a compact Riemann Surface except for a given point $p$.

I encounter the following problem. Given a compact Riemann surface $X$, and for any point $p \in X$, prove there exists a meromorphic function $f$ such that $f$ is holomorphic on $X\setminus \{p\}$. ...
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Bijectivity of $\mathbb{P}^1 \simeq \mathbb{C}_\infty$ using charts.

So I have to show $\mathbb{P}^1 \simeq \mathbb{C}_\infty$ using their charts. Let $F: \mathbb{P}^1 \rightarrow \mathbb{C}_\infty$ be defined via $[z:w]$ $\mapsto$ $(\frac{\mathrm{Re}z \overline{w}}{\...
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Small path enclosing a point $p$

Can anyone help me see why this statement is true Let $p$ be a point of a Riemann surface $X$, and let $S$ be a subset of $X$ whose closure does not contain $p$, then theres is a closed path $\...
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28 views

correspondence between meromorphic functions on complex torus vs. $\mathbb{C}$ itself

I have an integration to do along the path which is the border of a parallelogram, namely, I need to show using integration the sum of the orders on the complex torus is $0$. I reparameterized it ...

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