Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [riemann-surfaces]

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

52
votes
3answers
5k views

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 ...
52
votes
2answers
4k views

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 ...
29
votes
4answers
4k views

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 ...
25
votes
1answer
1k views

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)=\...
24
votes
1answer
3k views

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 ...
22
votes
1answer
615 views

Is there a complex surface into which every Riemann surface embeds?

Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \to \mathbb{CP}^3$. It follows from the degree-...
21
votes
1answer
1k views

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?
21
votes
1answer
2k views

How to properly use GAGA correspondence

currently studying algebraic surfaces over the complex numbers. Before i did some algebraic geometry (I,II,start of III of Hartshorne) and a course on Riemann surfaces. Now i understood that by GAGA, ...
21
votes
2answers
2k views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus $$X=\{[x:y:z]\...
20
votes
0answers
1k views

Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
17
votes
1answer
2k views

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 ...
16
votes
3answers
1k 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 ...
16
votes
1answer
242 views

The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g-6$ geodesic length functions

Setting: It is well known that the Teichmüller space $T_{g,b}$ of an oriented Riemann surface $S_{g,b}$ of genus $g \geq 2$ with $b \geq 1$ boundary components (satisfying $2g + b \geq 3$) can be ...
15
votes
3answers
1k views

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 $\mathbb C$...
14
votes
1answer
9k views

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 ...
14
votes
2answers
4k views

$\mathbb{C}\mathbb{P}^1$ 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 ...
14
votes
2answers
3k views

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?...
13
votes
1answer
1k views

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.
13
votes
2answers
2k views

Calculating monodromy

I'm right now learning about Monodromy from self-studying Rick Miranda's fantastic book "Algebraic Curves and Riemann surfaces". Today, I read about monodromy, and the monodromy representation of a ...
13
votes
1answer
1k views

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-...
12
votes
3answers
3k views

Differential forms

I failed to understand the definition of holomorphic $1$-form on Riemann surfaces. Can one explain it here? I saw two definitions in the books of Miranda and Farkas-Kra. Definition 1.: Suppose that $...
12
votes
2answers
554 views

Fibres in algebraic geometry: multiplicity

Currently I am studying varieties over $\mathbb{C}$ and I know some scheme theory. My professor mentioned the other day that given a morphism of varieties over an alg. closed field $k$: $f: X \...
12
votes
2answers
339 views

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 ...
12
votes
2answers
3k views

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 ...
12
votes
1answer
391 views

The Wronskian of holomorphic differentials as a q-differential

I just went through a proof of the counting of Weierstrass points on a Riemann surface (References: Reyssat, Quelques aspects des surfaces de Riemann and Farkas & Kra, Riemann Surfaces) that says ...
11
votes
1answer
1k views

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 ...
11
votes
1answer
205 views

Looking for elementary proof that irreducible/smooth curve in $\mathbb C^2$ is connected in Euclidean topology of $\mathbb C^2$

Let $f(X,Y)\in \mathbb C[X,Y]$ be an irreducible polynomial. I know that the zero set of $f$ , $V(f):=\{(a,b)\in \mathbb C^2 : f(a,b)=0\}$ is connected in the usual Euclidean topology of $\mathbb C^2$ ...
11
votes
1answer
219 views

Constructing one-forms on a Riemann surface using the uniformization theorem

I found this statement about proving the existence of one-forms on a Riemann surface in an answer on MathOverflow: This deep fact is essentially the same as the uniformization theorem. The ...
11
votes
0answers
205 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 $\tau$-structure,...
10
votes
3answers
467 views

Is every algebraic curve birational to a planar curve

Let $X$ be an algebraic curve over an algebraically closed field $k$. Does there exist a polynomial $f\in k[x,y]$ such that $X$ is birational to the curve $\{f(x,y)=0\}$? I think I can prove this ...
10
votes
2answers
1k views

About Gauss-Bonnet Theorem

The Gauss–Bonnet theorem say that: If $\Sigma \subset M =\mathbb{R}^3$ is a compact 2-dimensional Riemannian manifold without boundary, then $$ \int_{\Sigma} K = 2\pi\chi_{\Sigma}$$ where $K$ is the ...
10
votes
1answer
2k views

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 ...
10
votes
2answers
2k views

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-...
10
votes
1answer
319 views

Reasons for defining sheaves of holomorphic and meromorphic functions on complex manifolds

I am hoping this question is sensible and non-trivial. I am learning algebraic geometry at the moment, and have taken a strong liking to it. Unfortunately my complex analysis is weaker and I only know ...
10
votes
1answer
586 views

First derivative of the Weierstrass $\wp$ function as a function on $\mathbb{C}/\Lambda$

I am currently trying to prove various facts about $\wp'$, considered as a meromorphic map from $\mathbb{C}/\Lambda\to\mathbb{C}$, where $$\wp'(z) = -2\sum_{w\in\Lambda}\frac{1}{(z-w)^3}.$$ In ...
10
votes
1answer
331 views

constant-curvature Riemannian metric for Bring's surface

There is a well-known and very symmetric space that is called either "Bring's curve" or "Bring's surface", depending upon the context. (Bring was a Swedish mathematician in the 18th century.) Let's ...
10
votes
1answer
505 views

Basis of cohomology of curve

Let $R$ be a Riemann surface of genus g and $p\in R$ a point. I'm searching for a way to compute linearly independent differential 1-forms on $R$ which: are closed and not exact, not holomorphic, ...
10
votes
1answer
168 views

Is there any obstruction other than Riemann-Hurwitz to the existence of covers of Riemann surfaces?

Suppose $X$ is a genus $g$ Riemann surface, and $h,d,e_i$ are positive integers such that $2-2g = d(2-2h) + \sum (e_i-1)$. Is there necessarily a Riemann surface $Y$ with a map $f: Y \rightarrow X$ ...
10
votes
0answers
150 views

Geometric meaning of an integral on a compact Riemann surface

Let $X$ be a compact Riemann surface and fix a volume form $\Omega$ on $X$ such that $\int_X\Omega=1$. Now let's fix a function $g:U\subset X\to\mathbb R$ on $X$ with the following properties: $U=X\...
9
votes
1answer
383 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 ...
9
votes
3answers
288 views

Is the curve $ z = e^{i\theta}\left(\frac{7}{8} + \frac{1}{4} e^{6i\theta}\right) $ algebraic?

Is this spirograph curve algebraic? I an only write it in polar coordinates: $$ z = e^{i\theta}\left(\frac{7}{8} + \frac{1}{4} e^{6i\theta}\right) $$ and here is a picture. It is a six-sided rose-...
9
votes
3answers
2k views

Is $\sqrt{z}$ a meromorphic function?

The literature seems rather coy on this point. While $\sqrt{z}$ is not meromorphic on the complex plane $\mathbb{C}$, can it be regarded as globally meromorphic on the appropriate Riemann surface (...
9
votes
2answers
1k views

Historical basis and mathematical significance of Riemann surfaces

It is written in Riemann Surfaces (Oxford Graduate Texts in Mathematics) by Simon Donaldson, that: "[t]he theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination ...
9
votes
1answer
165 views

The $j$-invariant of my donut

I have a donut. Its boundary is a two-dimensional surface embedded in three-dimensional space, and surely is homeomorphic to a torus. If we fix a Riemannian metric on the space, it induces a two-...
9
votes
1answer
447 views

Complex structure on the Jacobian of a Riemann surface

Let $X$ be a fixed smooth, connected, compact Riemann surface of genus $g$. The Jacobian variety $\mbox{Jac}(X)$, which parametrises isomorphism classes of holomorphic degree $0$ line bundles on $X$, ...
9
votes
1answer
338 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, ...
9
votes
2answers
1k views

Explicit computation of the Hodge codifferential

Question I'm given a Laplacian $\Delta_n=-4y^2 \cdot \frac{\partial^2}{\partial\bar{z} \partial z} + 4 iny \cdot \frac{\partial}{\partial\bar{z}}$, and I want it to be the Laplace operator associated ...
9
votes
1answer
779 views

Example of integration over path on Riemann surface

Let $X$ be a Riemann surface $$ X = \left\{ (z,w) \in \mathbb{C}^2 \mid z^3 + w^3 = 1 \right\}. $$ Then we have $z^2 dz + w^2 dw = 0$ and we can define a holomorphic form $\omega$ on $X$ by $$ \...
9
votes
1answer
699 views

On the automorphisms of the Klein Quartic

I am trying to solve a problem from Miranda's book, Algebraic Curves and Riemann Surfaces. On page 84, problem K gives the Klein curve $X$ as a smooth projective plane curve defined by the equation $...
8
votes
3answers
3k views

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 ...