# Tagged Questions

22 views

### What is the definition of Riemann surface of an algebraic function?

What does it mean by the Riemann Surface of a function $y=\sqrt{x^3}$? I saw how to use the cut and glue method to obtain a sphere where $y=\sqrt{x}$ can be defined. But I was not clear in what sense ...
42 views

### Simple Branched covering over sphere.

A simple branched covering is a branched covering with branching points of degree at most 2, in some context, it is also required to have at most one branching point in each fiber. My question is ...
88 views

### Why is the projection map proper?

It might be a silly question. I got stuck there. For the context, see S.K.Donaldson's Riemann Surfaces, chapter 4, section 4.2.3. Suppose $P(z,w)=a_0(z)+a_1(z)w+\dotsb+a_n(z)w^n\in\mathbb C[z,w]$ is ...
73 views

Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
72 views

### Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
38 views

### Generators of the first singular homology group of a Riemann surface

Let $X$ be a Riemann surface embedded in $\mathbb C^2$ with coordinates $(z,w)$ and let $\pi_z \colon X \to \mathbb C$ be the projection on first coordinate with property that that for cofinite number ...
49 views

### Dimension of a meromorphic differentials space

What is the dimension $d_{ \large k,n}$ of the space of the degree $k$ meromorphic differentials on the sphere with fixed residues ($\alpha_i$) at $n$ points $z_i$ ? The question is asked in this ...
58 views

### Existence of a holomorphic map from Riemann Surface to an algebraic curve .

Let $C$ be an algebraic curve in $\mathbb P^2( \mathbb C)$ with singular points $p_i : \{1 \le i \le n \}$ . Then there exists a holomorphic map $\Phi : S \to C$ , where $S$ is a Riemann surface. ...
61 views

### $C^{\infty}$ 1-form on a Riemann surface is unique.

Let $X$ be a Riemann surface and $\mathcal{A}$ be a complex atlas on $X$. Suppose that $C^{\infty}$ 1-forms are given for each chart of $\mathcal{A}$, which transform to each other on their common ...
251 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 ...
213 views

### Modular functions of weight zero

The following question was suggested by Sasha's answer to the following question : Is the derivative of a modular function a modular function . Question. What are the modular functions with respect ...
362 views

### An argument on page 62 of Griffith's book, “Introduction to Algebraic Curves”

I am a bit confused about some of the things that Griffiths says on page 62 of his book, Introduction to Algebraic Curves. I am not sure how I can reproduce the text here. I can see that GoogleBooks ...
Let $V$ be an irreducible complex plane algebraic curve, $V=V(f)$, and let $\mathcal{O}_p$ be the local ring of holomorphic functions defined in some neighborhood of $p$. If $p=(0,0)$ is a smooth ...
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface \$V_{f, \kappa} = ...