# Tagged Questions

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### zeros and poles of meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$

I'm a bit confused regarding the following example: it is stated that $$\frac{x^3+y^3+z^3}{x^2yz}$$ is a meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$, and then one is asked to find poles ...
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### some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromorÂ­phic functions on $U$ is a field. Is it true ...
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### Non-hyperelliptic Riemann surface

Let $S$ be the Riemann surface of the plane algebraic curve $XYZ^3+X^5+Y^5 = 0$. How can I prove that $S$ isn't a hyperelliptic Riemann surface?
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### Mapping unit disc onto upper half plane

How can I map the unit disc onto the upper half plane? I tried mapping $(1,i,-1)\rightarrow(1,0,\infty)$ using cross-ratio: $z\rightarrow \frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, but didn't give ...
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### Show that a complex expression is smaller than one

This is the question I'm stumbling with: When $|\alpha| < 1$ and $|\beta| < 1$, show that: $$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$ The chapter that contains ...
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### The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle

I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I've done is I considered the $n$th roots of unity and said ...
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### Continuing a Divisor

I need some help, to find the strategy for solving the following problem: Given an analytic surface $S^0$, its compacitfication $S$ and a horizontal Divisor $D^0$ on $S^0$, I have to continue $D^0$ to ...
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### Geometrically interpret the set $\{z \in \Bbb C :$ $|z+a|+|z-a| \leq 2b,$ $b\in\Bbb R^+,$ $|a|<b\}.$

Geometrically interpret and determine the following set of complex numbers: $$\{z \in \Bbb C : |z+a|+|z-a| \leq 2b , b\in\Bbb R^+,|a|<b\}.$$ I understand it means the sum of distance between ...
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### A complex function

I just need some help with the penultimate question of my coursework: Let $w=f(z)=\coth(z/2)$. Show that $w=f(z)=h(C) = (C+1)/(C-1)$ where $C=g(z)=e^z$. Find the image of the given points, ...
### How to construct a simple complex torus of dimension $\geq 2$?
This is an exercise in the first chapter of "Complex Abelian Varieties" by Christina Birkenhake and Herbert Lange. As the title suggests it asks for an example of a complex torus of dimension $\geq 2$ ...