1
vote
0answers
18 views

About the equivalence between smooth projective curves and compact Riemann surfaces

We know that there exists an equivalence of categories between compact Riemann surfaces and smooth projective curves (over $\mathbb C$). But the Riemann surfaces are connected by definition, so I ...
2
votes
1answer
51 views

Prove that $H^1(\mathcal{M}^*)=0$.

Let $X$ be a compact Riemann surface. For an open set $U$, let $\mathcal{M}^*(U)$ be the multiplicative group of nonzero meromorphic functions on $U$ ("nonzero" meaning "not identically zero"). This ...
2
votes
0answers
32 views

Mobius transformation of Algebraic curve

I am working on the uniformization of algebraic curve problem. Currently, my adviser gave me a question about build a Mobius transformation between algebraic curves, and then lift it to the Rimeann ...
0
votes
1answer
25 views

Covering of Projective line

This is an exercise given during the course in Riemann Surfaces that I attended this year. Let $X$ be a compact Riemann Surface that is a degree $3$ cover of $\mathbb{P}^1(\mathbb{C})$ given by ...
0
votes
0answers
53 views

How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
1
vote
0answers
36 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 ...
0
votes
0answers
29 views

Function fields for genus 2 curves

Let's say you were given a genus one algebraic curve by the equation $y^2 = (x-a)(x-b)(x-c)$ and you wanted to parametrize it. We could go ahead and convert it to Weierstrass form: $y^2 = 4t^3 - g_2t ...
1
vote
1answer
85 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 ...
3
votes
0answers
42 views

Automorphisms of rational curves

Let $X$ be a non-empty open subscheme of $\mathbb P^1_{\mathbb C}$. What is the automorphism group of $X$ in terms of PGL$_n(\mathbb C)$ and the points on the boundary?
2
votes
1answer
58 views

Existence of $g_{d}^{r}$ implies existence of which $g_{d'}^{r'}$'s?

Suppose I have a Riemann surface with a $g_{d}^{r}$. I am wondering what $g_{d'}^{r'}$'s exist for sure. For instance, since $h^{0}\left(L\left(-p \right)\right)$ is either $h^{0}\left(L\right)$ in ...
3
votes
1answer
63 views

Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem: If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most $3$ ...
1
vote
1answer
103 views

When branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$.

I'm working on the following problem for several days without finding any solution: Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb ...
3
votes
1answer
61 views

$d$-branched coverings and finite morphisms of degree $d$

Consider a smooth projective curve $X$ over $\mathbb C$ (so $X$ is a projective $\mathbb C$-scheme, integral, of finite type....), and let $t: X\longrightarrow\mathbb P^1_{\mathbb ...
8
votes
3answers
201 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 ...
5
votes
1answer
139 views

How to compute this Riemann surface?

This question is related to other more general question that I asked Computing Riemann surfaces of a given algebraic function. By the way, I've found an approaching in Markushevich's book that ...
1
vote
0answers
38 views

Pushforward of differentials (?) and Abel Jacobi map on principal divisors

Let $X$ be a complete one-dimensional curve over $\Bbb C$ (please add any hypotheses in case I forgot them). Then we have the Abel-Jacobi map $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
3
votes
0answers
70 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 ...
0
votes
0answers
29 views

Questions on linear subspace of a projective space

I am a bit confused by the definition of the linear subspace of a projective space. It says in a book "Algebraic Geometry: A first course" by Joe Harris on page 5 that An inclusion of subspace ...
8
votes
1answer
161 views

Threefold category equivalence: algebraic curves, Riemann surfaces and fields of transcendence degree 1

According to this article, The category of curves over the complex numbers is equivalent to two other categories: Riemann surfaces and fields of transcendence degree 1 over $\mathbb{C}$. But I ...
2
votes
1answer
96 views

Linear Equivalence of Divisors on Projective Plane Cubic

I'm self-studying Miranda's Algebraic Curves and Riemann Surfaces and am uncertain of how I'm supposed to solve problem V.2c on linearly equivalent divisors: Let $X$ be the projective plane cubic ...
1
vote
1answer
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 ...
3
votes
1answer
81 views

Divisors on a complex torus

I'm asked to prove the following fact: On a complex torus $X$ every canonical divisor is principal and vice-versa. At this moment I know only the basic properties of divisors and that, if $K$ is a ...
2
votes
1answer
109 views

Plane algebraic curves in $\mathbb C^2$ are connected in the analytic topology.

Is there a "simple" proof, not involving much tools of Algebraic Geometry, to the fact that every irreducible affine curve $C=\{(z,w)\in\mathbb C^2\,:\, F(z,w)=0\}$ (where $F\in\mathbb C[X,Y]$ is ...
2
votes
1answer
125 views

About Linear Systems on Curves.

Let $C$ be a smooth irreducible (complex) curve of genus $g\geq2$. The gonality of $C$ is defined as the minimum degree of surjective morphisms $C\rightarrow\Bbb{P}^1$. So $C$ has gonality $d$ if it ...
4
votes
1answer
48 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 ...
1
vote
3answers
189 views

Compact Riemann surfaces are projective varieties.

We know that every compact Riemann surface is a complex compact manifold of dimention one. But why every compact Riemann surface is a projective variety?
11
votes
1answer
206 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?
4
votes
1answer
123 views

Fundamental group of a complex algebraic curve residually finite?

Is the analytic fundamental group of a smooth complex algebraic curve (considered as a Riemann surface) residually finite?
7
votes
1answer
155 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 ...
2
votes
0answers
51 views

Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
2
votes
1answer
57 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. ...
7
votes
1answer
343 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 ...
2
votes
1answer
94 views

What's the sense in a Hyperelliptic Riemann Surface?

Can someone explain me, possibly using some very intuitive ideas, of what kind of object a hyperelliptic Riemann Surface is? What's the goal of constructing it (my lecture on is was based in Miranda's ...
2
votes
0answers
83 views

Computing wedge product of two 1-forms.

Let $L$ be a lattice in $\mathbb{C}$ and let $\pi :\mathbb{C}\to X=\mathbb{C}/L$ be a quotient map. Show that the local formula $dz$ in every chart of $\mathbb{C}/L$ is a well-defined holomorphic ...
2
votes
1answer
146 views

Computing the intersection divisors of a smooth projective cubic curve.

Let $X$ be the smooth projective plane cubic curve defined by $y^2z=x^3-xz^2.$ Compute the intersection divisors of the lines defined by $x=0,y=0,$ and $z=0$ with $X$. Here is an idea: Any point ...
3
votes
1answer
200 views

Compute the principal divisors of a hyperlelliptic surface.

Let $X$ be the hyperelliptic surface defined by $y^2 = x^5-x.$ Note that $x$ and $y$ are meromorphic functions on $X.$ Compute the principal divisors div($x$) and div($y$). We have the ...
4
votes
1answer
387 views

Computing the divisors of a meromorphic function defined by a hyperelliptic curve.

Let $X$ be a hyperelliptic curve defined by $y^2=h(x).$ Let $\pi : X\to \mathbb{P}^1$ be the double covering map sending $(x,y)$ to $x$. Let $\omega=\pi^{*}(dx/h(x)).$ Compute div$(\omega)$. I ...
3
votes
1answer
81 views

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
5
votes
0answers
121 views

Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
1
vote
0answers
38 views

Is $M_g$ a subvariety of $M_{h}$ for some $h>g$

Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$? If the answer is not known, ...
2
votes
1answer
260 views

what is genus of complete intersection for: $F_1 = x_0 x_3 - x_1 x_2 , F_2 = x_0^2 + x_1^2 + x_2^2 + x_3^2$

In the below problem I want to answer second part: Show that the curve in $\mathbb{P}^3$ defined by the two equations $x_0 x_3 = x_1 x_2$ and $x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0$ is a smooth complete ...
2
votes
0answers
90 views

Are there infinitely many rational functions of bounded degree and given ramification

It is well known that the set of branched covers $X\to \mathbf{P}^1(\mathbf{C})$ of bounded degree and given branch locus is finite (up to isomorphism). Edit. The branch locus $B$ of $f:X\to ...
2
votes
1answer
82 views

the divisor of a rational function

Is the section $df$ associated to a rational function $f$ on a curve $X$ a global section of the canonical sheaf $\omega_X$? I know its zeroes are the ramification points, but does it have poles?
2
votes
2answers
67 views

Factorizing rational functions of curves

Let $f:X\to \mathbf{P}^1$ be a rational function of degree $d\geq 2$ on a curve $X$. Let $n\geq 2$ be a divisor of $d$. Does there exist a curve $Y$ with a rational function $g:Y\to \mathbf{P}^1$ of ...
3
votes
1answer
78 views

Does de Franchis' theorem hold over any base field

Let $k$ be a field and let $X$ be a hyperbolic curve over $k$. Then, there are only finitely many hyperbolic curves $Y$ over $k$ dominated by $X$. I know this statement holds over $k=\mathbf{C}$. In ...
0
votes
1answer
100 views

Representing a curve as a plane curve in different ways

Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$. I know that $X$ has a plane model. More ...
7
votes
1answer
136 views

The number of curves of given genus over a field

Let $k$ be a field. Let $g\geq 0$ be an integer. I have an elementary question. Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of ...
3
votes
1answer
103 views

How can function fields have different degrees over the projective line

I'm confused. Let $X$ be a curve over a field $k$. Let $K= K(X)$ be its function field. Then, $K(X)$ is a field. Each non-constant morphism $f:X\to \mathbf{P}^1_k$ gives a field extension ...
4
votes
2answers
191 views

Genus of curves embedded into some projective space

The Plucker formula allows one to calculate the genus of a curve embedded into $\mathbf{P}^2$. Does there exist a "higher-dimensional" analogue of the Plucker formula? More precisely, let ...
2
votes
2answers
198 views

Rational points on singular curves and their normalization

Let $X$ be a curve over a field $k$. Assume that $X$ is geometrically connected, geometrically reduced and stable. Let $Y\to X$ be the normalization. Is $Y(k) = X(k)$?