# Tagged Questions

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### 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 ...
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### Tracing down (or pushing forward) 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 $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
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### 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: $$\chi(D)=\deg D+1-g(C).$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### On the original Riemann-Roch theorem

I think Riemann first stated and proved a part of the Rieman-Roch theorem on a compact Riemann surface. And later Roch supplemented it. I wonder what the original statements of the R-R theorem by ...
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### Is the divisor of a meromorphic function “actually” a “divisor-valued potential”?

Obviously this is an open-ended question, so I'll be happy to try and clarify anything about it that doesn't make sense. Let $X$ be a Riemann surface and $f$ be a meromorphic function on $X$. Of ...
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### Visualize a projective curve $X^3+Y^3=Z^3$ in $P_2(C)$ as a torus

Let $P_2(C)$ be the 2 dimensional complex projective space, I want to prove that the projective curve defined by $M=\{[X,Y,Z]\in P_2(C)|X^3+Y^3=Z^3\}$ is a torus. I know that there is a theorem saying ...
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### Möbius transforms on the projective line

Let's suppose that I have a mapping $$\phi:\mathbb{P}^1\rightarrow\mathbb{P}^1$$ of degree two. How can I use Möbius transforms to write this map as $$\phi([x:y])=\frac{y^2}{x^2}?$$ I already ...
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### Immediate consequence of Riemann-Roch

Let $X$ be an algebraic curve, $D$ a divisor and $\mathscr{O}(D)$ the line bundle associated to $D$ in the canonical way. The following implication should follow immediately from the Riemann Roch ...
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### When a divisor on an algebraic curve is canonical

I am trying to solve this: If $D$ is a divisor over a algebraic curve X of genus $g$ such that $\deg(D)=2g-2$ and $\dim L(D)=g$, then $D$ is a canonical divisor Using Riemann-Roch theorem, ...
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### On the definition of divisors in Riemann Surfaces

The sum notation for a Divisor $D$ in a Riemann Surface $X$ (as in Miranda's "Algebraic Curves and Riemann Surfaces") is $$D=\sum_{p\in X} D(p)\cdot p$$ That is, $D$ assumes the value $D(p)$ at $p$. ...
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### Algebraic analogue of maximum modulus principle applied to Riemann surface.

Let $X$ be an abstract curve in the following sense: $X$ is a scheme, proper over $k$ which is noetherian, integral, dimension 1, and normal. The important thing to point out is that I am not assuming ...
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### Partition of Unity for the Divisor Sheaf

Recall that given a Riemann Surface $X$, the divisor sheaf is the sheaf ${\cal D}$ which assigns to each open set $U$ the collection of maps $\phi:U \to \mathbb{Z}$ such that $\phi(p)=0$ for all but ...
How do I compactify the curve $Q(x,y)=0$ in $\mathbb{P}^1\times\mathbb{P}^1$ where $Q$ is a polynomial ?
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 ...