1
vote
0answers
38 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
2
votes
1answer
31 views

Holomorphic map or Riemann suface into projective space, Miranda's book

I have the following question after reading Chapter V, prop. 4.3 of Miranda's book Algebraic Curves and Riemann Surfaces. The setting is as follows: we have a Riemann surface $X$ and a holomorphic ...
1
vote
0answers
15 views

Finding an algebraic equation given divisors

I'm trying to find an algebraic curve that represents a specific Riemann surface and my question goes like this: Given divisors $(\omega_1) = P_1 + 5 P_2 + 2 P_3,$ $(\omega_2) = 5 P_1 + P_2 + 2 P_3,$ ...
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 ...
1
vote
1answer
80 views

Holomorphic line bundle with degree zero

I'm studying algebraic geometry and I need some help to understand the Riemann-Roch theorem. Let us consider a holomorphic line bundle $\xi$ over a Riemann surface $X$. The unique invariant of a ...
4
votes
0answers
56 views

Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism.

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
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
2answers
177 views

1-form on Riemann Surface

Good evening, I can not prove the following result: Let $\omega $ be a meromorphic 1-form on $ \mathbb {C} _ {\infty} = \mathbb {C} \cup \infty $ such that $ \omega_{|\mathbb{C}} = f (z) dz $. Show ...
1
vote
1answer
38 views

Base-point-free linear systems (elementary?) property

I'm having troubles solving exercise K on page 167 of the book "Algebraic curves and Riemann surfaces" of Miranda. The question is the following one : Let Q be a base-point-free linear system, let ...
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?
3
votes
1answer
72 views

Galois extension and morphism of curves

Let $\phi: C \rightarrow \mathbb P^1$ a morphism (over a field of characteristic 0) from a rational curve $C$ to $\mathbb P^1$ of degree 3. By the Riemann-Hurwitz formula the degree of the ...
2
votes
1answer
64 views

Universal property of the Abel map

In the book Algebraic Geometry I edited by Safarevich, the following universal property of the Jacobian variety of an algebraic curve is given page 158 (with no more details): The Abel mapping $a: ...
4
votes
0answers
57 views

Fermat Quartic Tiling

I have been reading about the Fermat quartic $Q \subset \mathbb{P}^{2}$, defined in homogeneous coordinates as $X^{4}+Y^{4}+Z^{4}=0$. This is the second most symmetric non-hyperelliptic surface of ...
0
votes
2answers
57 views

Topological space underlying this curve

I have to solve this exercise but I have really no clue even how to start with it: Identify the topological space underlying the cubic $Y^2Z=X^2(X-Z)$ in $\mathbb{PR}^2$. How does it fit with the ...
4
votes
0answers
52 views

Triple Cover of the Riemann Sphere

I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following: (i) The canonical embedding $\phi: X \rightarrow \mathbb{P}^{3}$ can be given in ...
2
votes
1answer
98 views

Proving the Existence of an Automorphism on $\mathbb{P}^{1}$

I recently came across the following problem while reading: Suppose that a compact Riemann surface $X$ has genus $g>1$. Let $\phi_{i}:X \rightarrow \mathbb{P}^{1}$ for $i=1,2$ be a pair of ...
4
votes
2answers
59 views

Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
5
votes
0answers
76 views

Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
5
votes
0answers
49 views

Mittag-Leffler Problem

We have: $X$ a compact Riemann surface defined by $y^{2}=1-x^{6}$ and $P=(0,1) \in X$ a point given in local coordinates $(x,y)$. Furthermore, we have a meromorphic function $f(x,y)=y/x$ such that $f ...
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 $3$ ...
3
votes
1answer
93 views

Weierstrass Point of a Riemann surface

I have that $X$ is a compact Riemann surface defined by the curve $y^{2}=1-x^{6}$ and a point $P=(0,1) \in X$ in the usual coordinates $(x,y)$. Ultimately, I want to solve a Mittag-Leffler problem on ...
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 ...
1
vote
0answers
38 views

Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$ This is the same as giving the (very ample) line bundle over the curve ...
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
105 views

Riemann-Roch theorem for singular curves

It might be a naive question, but I just realized I had not thought about this before. If $C$ is a smooth curve, for any line bundle $D$ we have the Riemann-Roch formula: $$\chi(D)=\deg D+1-g(C).$$ ...
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 ...
5
votes
1answer
89 views

Meromorphic functions a constant sheaf?

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic ...
0
votes
0answers
35 views

When is a map $\mathbb{CP}^1 \to \mathbb{CP}^2$ a holomorphic embedding?

Consider the map $\mathbb{CP}^1 \ni (u:v) \mapsto (x:y:z) \in \mathbb{CP}^2$ given by $$ x= u^2, \quad y=v^2, \quad z=uv.$$ Is it a holomorphic embedding? What is to be checked, perhaps via some ...
7
votes
1answer
207 views

Projective closure of an algebraic curve as a compactification of Riemann surface

Assume $f \in \mathbb{C}[x,y]$ a polynomial such that the affine algebraic curve $X=V(f)$ has no singular points. Then there is a natural structure of non-compact Riemann surface on $X$, which can be ...
1
vote
1answer
46 views

Computing the order of $dx$ at the infinity point of an elliptic curve

I am having trouble to figure out the order of the differential form $dx$ is on the infinity point $P=[0:1:0]$ of the elliptic curve $C$: $$y^2 = (x-e_1)(x-e_2)(x-e_3)$$ I want to compute this ...
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 ...
1
vote
2answers
76 views

Elliptic Curve and Conjugation

If I consider an elliptic curve $C$ as a Riemann surface cut out in $\mathbb{C}P^2$ by a homogenous cubic, and if that cubic is defined over $\mathbb{R}$, then I think we have a conjugation map ...
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 ...
5
votes
0answers
65 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
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 ...
3
votes
0answers
53 views

SU(2) Lefschetz decomposition of Jacobian of Riemann surfaces

Start with a (closed and oriented) Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in (the cohomology of) its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or ...
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 ...
3
votes
0answers
99 views

Reference about Moduli Spaces of Riemann surfaces

I'm looking for some (introductory, and in any case not too technical) reference (book, lecture notes, papers) regarding moduli spaces $\mathcal{M}_g$ and $\mathcal{M}_{g,n}$ of (punctured) Riemann ...
0
votes
0answers
22 views

definite integral of Abel-Jacobi map on Riemann surface

Suppose you're given the Riemann surface $$ 0= e^{-u}+e^{-v}+e^{u-v-t}+1,$$ where $u,v$ are complex variables. Can anyone explain what is the Abel-Jacobi map on this surface, what is its relation to ...
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?
0
votes
0answers
127 views

Action of the fundamental group on a Universal cover

Let $\pi: \tilde{X} \mapsto X$ an universal cover. I know that $\tilde{X}/Aut(\tilde{X},\pi) \simeq X$. Let $H \subset \pi_1(X,q)$ a subgroup of the fundamental group and consider the orbit space ...
1
vote
0answers
41 views

Group of covering transformations

The group of automorphisms of a covering $p: E \mapsto X$, to be denoted $Aut(E,p)$, is usually referred to as the group of covering transformations. If $p: E_1 \mapsto E_2$ is an isomorphism of ...
3
votes
1answer
109 views

Proper map on compact Riemann surface

i hope you can help me with a problem I discovered while dealing with the resolution of singularities for an algebraic curve in $\mathbb{P}^2$. A resolution means for me, that for an algebraic curve ...
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?