# Questions tagged [riemann-surfaces]

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

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### 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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### 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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### 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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### 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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### 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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### 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? ...