An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities ...

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1answer
130 views

Can a non-proper variety contain a proper curve

Let $f:X\to S$ be a finite type, separated but non-proper morphism of schemes. Can there be a projective curve $g:C\to S$ and a closed immersion $C\to X$ over $S$? Just to be clear: A projective ...
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1answer
135 views

Are elliptic curves also Galois covers of degree 3

Let E be an elliptic curve with equation $y^2=x^3+Ax+B$. The projection onto the $x$-coordinate is a Galois morphism of degree $2$. But what about the projection onto the $y$-coordinate? Is it ...
5
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1answer
262 views

Why is Klein's quartic curve not hyperelliptic

Let $X$ be Klein's quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$. How do I easily show that $X$ is not hyperelliptic? I can see that $X$ is of genus $3$ ...
2
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1answer
419 views

Parameterizing a rational curve

I'm having trouble finding a parameterization for the following curve: $x^4 - 2x^2yz + y^2z^2 - y^3z = 0$ taken to be a curve in $\mathbb{C}\mathbb{P}^2$. I followed the example on Wikipedia where ...
7
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1answer
191 views

Is the Fermat scheme $x^p+y^p=z^p$ always normal

Let $K$ be a number field with ring of integers $O_K$. Is the closed subscheme $X$ of $\mathbf{P}^2_{O_K}$ given by the homogeneous equation $x^p+y^p=z^p$ normal? I know that this is true if ...
3
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1answer
143 views

self-intersections in a product of two curves

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field and consider the intersection pairing on the surface $X \times X$. I remember hearing that $\Delta^2 = 2-2g$: how ...
2
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1answer
88 views

Residue map of a place

The following definition for a residue map is given in "Algebraic Curves over a Finite Field" by Hirschfeld, Korchmáros and Torres (page 265): "Let $\Sigma$ be a field of transcendence degree 1 over ...
3
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1answer
132 views

A field isomorphism related to polynomial rings and their field of fractions

There are 2 ways to approach function fields: the algebraic approach, i.e. looking at finite extensions of $K(s)$, where $s$ is transcendental. The other is geometric, i.e. considering functions over ...
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0answers
350 views

What is a global section, and why do we use cohomology?

I have a few questions about the use of cohomology. Firstly,we use cohomology to measure the obstruction of a section from a global section, so what can we do about a global section? I got very ...
3
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1answer
130 views

Can an algebraic group only have trivial elements over $k$

Let $G$ be an algebraic group over $k$ such that $G(k) = \{e\}$ is the trivial group. Does this imply that $G_{\overline{k}}$ is trivial? I think the answer is no. I think you can just take an ...
2
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0answers
134 views

Intersection of a cone $x^2+y^2-z^2$ and a generic plane in $\mathbb{RP}^3$

If we take the zero locus of $x^2+y^2-z^2$ to be our cone, I'd like to know how to go about finding the intersection of the cone and a generic plane $Ax+By+Cz+Dw=0$. The result will be a conic, but ...
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0answers
290 views

What is the equation for a cone in $\mathbb{RP}^3$?

The zero locus of $x^2+y^2-z^2$ is a cone in $\mathbb{R}^3$. What is the projective version of this cone? That is, what is the homogeneous polynomial whose zero locus is a cone in $\mathbb{RP}^3$? ...
3
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0answers
64 views

What applications does the theory of fibered surfaces have

Let $C$ be a smooth projective connected curve over $\mathbf{C}$. Let $X$ be a curve over the function field of $C$. Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ ...
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0answers
78 views

Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes

Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$. ...
6
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1answer
348 views

Intersection number & non-singular curves

I was looking at a proof of the following theorem... Let $S/\mathbb{C}$ be a smooth projective surface, let $C$ be a non-singular irreducible curve on $S$. Then for all $L \in \text{Pic}\,S$, the ...
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0answers
77 views

Is the degree of a Galois morphism bounded by $84(g-1)$

Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$. Assume $g=g(X) \geq 2$. Is the degree of $X\to Y$ bounded by $84(g-1)$? I ...
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1answer
147 views

Does the absolute Galois group act on the moduli space of curves

Do elements of the absolute Galois group $G=\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ of $\mathbf{Q}$ induce automorphisms of $\mathbf{C}$ if we extend the morphisms trivially to the rest of ...
3
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1answer
103 views

For curves, is being defined over a number field invariant under birational equivalence

Suppose that a (connected) Riemann surface $X$ is birational to a Riemann surface $Y$ which can be defined (algebraically) over the field of algebraic numbers. Does this imply that $X$ itself can be ...
3
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2answers
169 views

Does there exist a finite morphism of algebraic curves such that…

Let $K\subset L$ be a finite field extension. Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$. Let $f:X\to Y$ be a finite morphism of curves over $L$. Assume that ...
3
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2answers
237 views

Are fundamental groups of Riemann surfaces always finitely generated

For any finite subset $B\subset \mathbf{P}^1$, the fundamental group of the Riemann surface $\mathbf{P}^1-B$ is finitely generated. Is this true if we replace $\mathbf P^1 $ by a higher genus compact ...
4
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1answer
123 views

Ramification of a prime in a Dedekind ring and curves

Let $\phi:C_1\to C_2$ be a nonconstant map of two smooth curves over some algebraically closed field $K$ and let $P\in C_1$. $\phi$ gives us an induced map of fields $\phi^*:K(C_2)\to K(C_1)$, ...
5
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1answer
182 views

Relationship between two distinct notions for divisors on curves

I've seen divisors on curves before, a few years ago in a course in algebraic geometry. Now I've come across them again, but they're somewhat more generalized. I was hoping someone could explain the ...
3
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1answer
85 views

Do gonal morphisms have non-trivial automorphisms

Let $X$ be a compact connected Riemann surface. Let $\pi:X\to \mathbf{P}^1$ be a gonal morphism, i.e., a morphism of minimal degree. Can $\pi$ have non-trivial automorphisms? (An automorphism of ...
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1answer
165 views

Galois covers of Riemann surfaces

Let $G$ be a finite abelian group, and $C$ a compact Riemann surface (algebraic curve) of genus $g$. I am interested in topological Galois $G$-covers $X \to C$, aka \'etale $G$-principal bundles over ...
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3answers
453 views

Is a cover Galois if and only if it is geometrically Galois

Let $K$ be a number field and let $\pi:X\to \mathbf{P}^1_K$ be a finite morphism, where $X$ is a smooth projective geometrically connected curve. Is $\pi$ a Galois cover if and only if the base ...
2
votes
1answer
70 views

Covers without automorphisms

Let $X\to \mathbf{P}^1$ be a branched cover of the complex projective line, where $X$ is a compact connected Riemann surface. Let $G=\mathrm{Aut}(Y/\mathbf{P}^1)$. Question 1. Could somebody provide ...
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0answers
140 views

What is the Hurwitz number of an elliptic curve

One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points. Suppose that $X$ is an elliptic ...
2
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1answer
67 views

What is the length of the following local ring

Let $f:Y\to X$ be a finite etale cover of smooth projective connected varieties. (Or, just a finite degree connected topological cover of connected Riemann surfaces.) Let $y\in Y$ and let $x=f(y)$. ...
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0answers
52 views

Why does a non-constant endomorphism of curves have isolated fixed points

Let $f:X\to Y$ be a non-constant morphism of smooth projective connected curves over $\mathbf{C}$ (or compact connected Riemann surfaces). Suppose that $X=Y$ and that $f$ is not the identity. Why ...
2
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1answer
111 views

Is the intersection of the diagonal with a graph always transverse in characteristic zero

Let X be a projective smooth connected curve over $\mathbf{C}$. Let $f:X\to X$ be a non-constant morphism. Is the intersection of the diagonal $\Delta_X$ and the graph $\Gamma_f$ on $X\times X$ ...
3
votes
1answer
120 views

when the curve $\mathbb{r=a\sin(b\theta)}$ is algebraic?

A need to show that the curve given in polar equation $\mathbb{r=a\sin(b\theta)}$ is an algebraic curve if $b=\frac{m}{n}$, $m,n\in \mathbb{N}^{*}$ and $(m,n)=1$. Also I am supposed to find the ...
5
votes
1answer
239 views

The normalization of the peculiar curve $x^p + y^p - (x+y)^p = 0$

Fix a prime number $p$. Consider the affine curve $C$ in $\mathbf{A}^2$ over a number field $K$ given by the equation $x^p+y^p - (x+y)^p =0$. Its Jacobi matrix is $(px^{p-1} -p(x+y)^{p-1} \ py^{p-1} ...
3
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0answers
171 views

Minimal resolution of singularities of Fermat curve

Fix a prime number $p$. Let $F$ be the geometrically connected Fermat curve of exponent $p$ over a number field $K$, i.e., $F$ is given by the equation $x^p+y^p = z^p$. Consider the same equation ...
3
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0answers
86 views

Find an explicit isomorphism from a curve of genre zero to the Riemann sphere

I can't figure out this exercise: i have this singular curve in $\mathbb{P}^2\mathbb{C}$ given by $\{[X,Y,Z]\in \mathbb{P}^2\mathbb{C}:X^2Y^2+Y^2Z^2+X^2Z^2=0\}$, I have shown that its ...
6
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1answer
175 views

Integral closure in the total ring of fractions

My question is linked with normalization of reduced algebraic curves that are not necessarily irreducible. Let $(A,\mathfrak{m})$ be a local reduced noetherian ring with Krull dimension $1$, let ...
4
votes
1answer
212 views

ample sheaf which is not very ample

I'm having trouble understanding a remark in Hartshorne: Let $X$ be the nonsingular projective cubic defined by $y^2z = x^3 - xz^2$ and put $P_0 = (0,1,0)$. The claim is that $\mathscr{L}(P_0)$ is not ...
9
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1answer
184 views

What is the minimum background required to understand moduli of curves?

Recently I've coincidentally run into various relatives of the moduli stack $\mathcal{M}_g$ in several unrelated contexts. I tried reading Harris and Morrison's "Moduli of Curves," but it seems to ...
11
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2answers
257 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 ...
9
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1answer
439 views

Weil and Cartier divisors on a curve

I'm trying to understand the relationship between Weil divisors and Cartier divisors, and I would like to see why these are the same in the simple case where $X$ is a nonsingular projective curve over ...
2
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1answer
45 views

Are these sufficient conditions for point on an abstract curve to be regular?

Let $p\in{X}$ where $X$ is a curve-- here the definition of a curve is an integral, seperated, 1 dimensional scheme of finite type over a field $k$ (not necessarily algebraically closed). Moreover, ...
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0answers
105 views

Does there exist a number field with the following property

Let $\overline{\mathbf{Q}}\subset \mathbf{C}$ be the field of algebraic numbers. Does there exist a number field $K$ with the following property? There are embeddings $\sigma,\tau:K\to ...
4
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1answer
136 views

When is the canonical model of a curve nonsingular

Let $O$ be a Dedekind domain with fraction field $K$. Let $C$ be a smooth projective geometrically connected curve of genus $g>1$ over $K$. Let $p:X \to \mathrm{Spec} \ O $ be the canonical model ...
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1answer
68 views

Natural space to consider solution to polynomial equations

Why is the complex projective planes the most natural place to look to consider solutions of polynomial equation? Why is the complex plane $\mathbb{C}$ adequate for polynomial equations of one ...
7
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1answer
171 views

Twists of curves over number fields

Let $X$ be a curve over $\overline{\mathbf{Q}}$. I can prove that $X$ can be defined over some number field. (Take two equations defining $X$ in $\mathbf{P}^3$ and consider the number field ...
3
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1answer
159 views

Definition of Model of an Algebraic Curve

I'm reading the classical paper of Arakelov "Intersection Theory of Divisors on an Arithmetic Surface". At the very beginning he uses the notion of model of a curve. In specific we have a number ...
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1answer
242 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 ...
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0answers
379 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 ...
2
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0answers
132 views

exact sequence of sheaves

I'm starting with $X=\mathbb{P}^2(\mathbb{C})$ and a cubic curve $B \subset X$ and a flex $P$ on $B$ such that for a hyperplane section $H$ I have $3P \sim dH\vert_B$ (where $d \in \mathbb{N}$). With ...
0
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1answer
304 views

Irreducible conic implies that the underlying matrix is invertible

I guess that it is true that a conic (2nd degree homogeneous equation in complex variables) is irreducible (i.e can't be factorized over polynomials) if and only if the underlying matrix of ...
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0answers
206 views

Motivation and checking for points being in “general position”

I wanted to know the motivation and the calculation details of checking that a certain number of points are in "general position". Intuitively I was thinking that a set of points being in general ...