Questions tagged [riemann-surfaces]

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

Riemann surfaces are one-dimensional complex manifolds that are deformations of $\mathbb{C}.$

Riemann surfaces are orientable as a real manifold, and every simply connected Riemann surface is conformally equivalent to one of the following:

Elliptic (Positive curvature): The Riemann sphere (the complex plane with an extension to a point at infinity)

Parabolic (zero curvature): The complex plane

Hyperbolic (negative curvature): The open disk or upper half-plane

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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 I've seen, algebraic geometers in general) place…
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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 the following three notions of "degree"…
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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 prefer an explanation that avoids talking about…
Edward Hughes
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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 has any input on the matter. Next term I would…
Alex Youcis
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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 $$X=\{[x:y:z]\in\mathbb{P}^2:x^d+y^d+z^d=0\}$$ It is defined by…
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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 $E(\Lambda)=\mathbb C(\wp,\wp')$ and they "represent" all…
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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, a lot of results transfer from complex analytic…
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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?
Bruno Joyal
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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-genus formula that the same is not true if we…
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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 quotient space of $\mathbb{C}^3-\{0\}$ by the…
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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 $X$ be a smooth projective curve over $\mathbb{k}$. Let…
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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?). However, I'm looking for something a little…
Alex Petzke
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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 intimidated me when I first saw it as an undergrad, as the…
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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-theoretic topologist L.A. Tumarkin, who…
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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 are the differences in the approaches of these…
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