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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32 views

Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
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1answer
63 views

Show that a infinite discrete subset of $\mathbb{R}^n$ is not an algebraic set

I want to prove that a set which is discrete in $\mathbb{R}^n$ (with the euclidean topology) and infinite cannot be an algebraic set. How could I do it?
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2answers
37 views

Quartic in $\mathbb{P}^2_k$ are not hyperelliptic

Let fix an algebrically closed field $k$. It is easy to show that a curve of genus $3$ over $k$ is hyperelliptic or a quartic in $\mathbb{P}^2_k$. I have some difficulties to prove that there not ...
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1answer
79 views

Surjectivity of morphisms of smooth projective varieties

I have a question regarding a proof of the "surjectivity of morphisms of projective varieties" (a whole mouthfull). Though there are proofs using completeness of varieties, I am interested in an ...
2
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0answers
30 views

Explicit form of certain polynomials and intersection of curves

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ two effective divisors on $X$ intersecting at finitely many points. Is it true that if $C$ and $D$ intersect in ''low'' number ...
3
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26 views

Reference to an atlas of curves and surfaces?

I remember at more than one university math department there being a set of glass cabinets with a number of physical models of surfaces. They were all algebraic varieties on the reals (of limited ...
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1answer
78 views

Divisors of degree $2g-2$ on a hyperelliptic curve of genus $g$

Suppose I have a divisor $D$ of degree $2g-2$ on a hyperelliptic curve of genus $g$. Then I can prove that either a) $K_C\otimes\mathcal{O}(-D)=\mathcal{O}_C$, that is $K_C\cong \mathcal{O}(D)$, or ...
4
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1answer
61 views

Prove that a set in $\mathbb R^3$ is not an algebraic set

I want to prove that the set $\{(\cos(t),\sin(t),t)\in A^3(\mathbb R); t\in \mathbb R \}$ is not an algebraic set. I already proved that the set $\{(\sin(t),t)\in A^2(\mathbb R);t\in \mathbb R \}$ ...
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0answers
38 views

Continuous maps from an absolute Galois group

Let $\xi$ be a continuous homomorphism from an absolute Galois group $G_{\bar{K}/K}$ (Krull topology) to a finite abelian group $M$(discrete topology), where $K$ is a number field and $\bar{K}$ is its ...
8
votes
1answer
151 views

Characterization of the $m$-torsion points of an elliptic curve.

Let $(E,\mathcal{O})$ be the elliptic curve of equation $$ f=Y^{2}+a_{1}XY+a_{3}Y-X^{3}-a_{2}X^{2}-a_{4}X-a_{6}, $$ $\alpha:K(E)\rightarrow K(E)$ the derivation such that $$ \alpha(X)=\frac{\...
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78 views

Equation to Draw Curves with Saturation and Peak

I am looking for an equation to draw a graph like this: The curve should have a peak and saturation. Would you please let me know what is the equation that can generate similar curve ? Here is ...
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0answers
40 views

Global sections of symmetric product of curves

Let $C$ be a irreducible, smooth, projective curve over $\mathbb{C}$. Let $L$ be a globally generated line bundle over $C$. Let $h^0(C,L)=m.$ Consider the product $C \times C$. If $p_i:C \times C \...
6
votes
2answers
163 views

Normal bundle of twisted cubic.

Let $C$ be a twisted cubic in $\mathbb P^3$. I'd like to compute the splitting type of normal bundle $N_{C/\mathbb P^3}$? I understood that $T_{\mathbb P^3}|_C=\mathcal O(4)^{\oplus 3}.$ So we have an ...
3
votes
2answers
54 views

Computing $l(D)$ for certain divisor.

Let $C$ be a smooth projective curve of genus $g=2$. I want to prove that there exist $P,Q\in C$ such that $$ l(P+Q)=2. $$ I know that if $D\in Div(C)$, and $x\in C$, then $$ l(D)\leq l(D+x)\leq l(D)+...
3
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71 views

The Picard group of an Elliptic Curve

Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$ I understand ...
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70 views

Genus-Degree formula gives the wrong answer: ordinary points?

I'm trying to compute the genus of the normalization of the curve: $y^5=x(x-1)(x-2)$ Now I calculate the ramification points of the projection x: they are $(0,0),(1,0),(2,0)$ and they are of ...
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0answers
25 views

Is this the correct way to compute the blow up of a curve

I'm trying to calculate the blowup of the curve $y^5=z^2-3z^3+2z^4$ at $(0,0)$ We have the relation $Ay=Bz$, now I split it into two charts: The first chart$(y,a=A/B)$: $y^5=a^2y^2-3a^3y^3+2a^4y^2-y^...
3
votes
1answer
90 views

Endomorphism ring of Drinfeld modules.

Let $\mathcal{X}$ be a smooth geometrically irreducible projective curve over $\mathbb{F}_q$. Fix a closed point $\infty\in \mathcal{X}(\bar{\mathbb{F}_q})$. Let $K$ be the function field of $\mathcal{...
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1answer
39 views

Smoothing transverse self-intersection

Let $S$ be a complex surface, and let $C \subset S$ be an immersed complex curve with a transverse self-intersection at point $P$. Let $\tilde{C}$ be a curve obtained from $C$ by smoothing the ...
4
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43 views

Which curves have reflexive structure sheaf?

Let $C$ be a curve, i.e. a purely one-dimensional scheme, embedded in a smooth projective threefold $X$. For a coherent sheaf $E$ of codimension $c$ on $X$, let $E^D=\mathscr Ext_X^c(E,\omega_X)$ be ...
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1answer
54 views

Cohomology of rational quartic in $\mathbb{P}^3$

I have to do this exercise. Let $X\subseteq\mathbb{P}^3$ a rational curve of degree $4$. Show that $$H^1(\mathcal{O}_X(1))=0=H^1(\mathcal{I}_X(2))$$ I tried to look at $X$ as closed immersion by ...
2
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1answer
89 views

Is there something similar to $\mathbb{R}^2$ for elliptic curve point representation?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ and denote with $E(\mathbb{F}_p)$ its set of points over $\mathbb{F}_p$. Consider a coordinate system in $\mathbb{R}^2$. Every point is ...
2
votes
2answers
125 views

The equation of a jelly bean curve?

What is the equation of a curve with jelly bean shape? I have found a quartic equation for bean shaped curves, but nothing for jelly beans. If somebody doesn't know the shape, here is a link: jelly ...
3
votes
1answer
63 views

Computing cohomology over projective curve in $\mathbf{P}^3$

Let be $k$ an algebraically closed field and Let be $X\subseteq \mathbf{P}^3:=\mathbf{P}_k^3$ a smooth, irreducible curve that is not contained in any hyperplane. Let's call $d=\deg(X)$. A well known ...
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1answer
13 views

Prove that the tangent at $P$ intersects $C$ twice at $P$ and once at $4P$; the tangent at $5P$ intersects $C$ twice at $5P$ and once at $2P$.

This is a problem from Conics and Cubics by Bix. Please help me answer this one. Let $C$ be a nonsingular, irreducible cubic with a flex $O$. Add points (commutative) of $C$ with respect to $O$ as ...
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0answers
18 views

An application of the Max Noether's theorem

I'm reading chapter IV of Robert J Walker's book 'algebraic curves'. The last section of this chapter is about Max noether's AG+BF theorem. I am stuck on an exercise in this section. The exercise ...
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1answer
18 views

Prove that $2P$ is a flex of $C$ collinear with $P$ and $3P$ and $4P$ is a flex of $C$ collinear with $5P$ and $3P$.

I encountered this problem from Conics and Cubics by Bix. Please help me answer this. Let $C$ be a nonsingular, irreducible cubic with a flex $O$. Add points (commutative) of $C$ with respect to $O$ ...
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vote
1answer
46 views

Show that the irreducible quartic projective curve is rational

How can I show that the irreducible quartic curve $\Gamma=V_+((x^2-z^2)^2-y^2(2yz+3z^2))$ on $P^2(\mathbb{C})$ is rational by considering the family of conics through the double points $(1:0:1), (-1:0:...
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vote
0answers
30 views

Solving the curve equation for logarithmic decay using two anchor points.

I would like to have an adaptable logarithmic curve equation that I can then find y for any value of x. I have two points (x1,y1) and (x2,y2). My data requires constant decay (financial discounting ...
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0answers
45 views

Relationship between discriminants and smoothness of curves

My understanding of the use of the discriminant in elliptic curve theory is to test whether an elliptic curve in Weierstrass normal form over a field not of characteristic either 2 or 3, $y^{2} = x^{3}...
3
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0answers
72 views

Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity. What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are ...
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votes
1answer
30 views

How can I describe the intersection between a circle and a curve?

I have a curve C and a point x in the curve. At the point x, I draw a circle B with radius r and centered at point x. That circle B will segment/intersect (with) the curve C as red sub-curve line. I ...
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29 views

Resultants on Projective Curves

The two curves $$F=(X^2+Y^2)^2+3X^2YZ-Y^3Z$$ and $$G=(X^2+Y^2)^3-4X^2Y^2Z^2$$ on $P^2(\mathbb{C})$contain the point $(0:0:1)$ as their point of intersection. Therefore, the resultant with respect to $...
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0answers
6 views

does the $\zeta_K$ function of a function field determine the genus of that function field?

Let $K_1$ and $K_2$ both be function fields over a finite field (or algebraic curves, if you like) with zeta functions $\zeta_{K_1}$ and $\zeta_{K_2}$. Say that $\zeta_{K_1} = \zeta_{K_2}$ - so that ...
0
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0answers
29 views

Rational Parametrization of Projective Curves

I wish to show that the curve on $P^2(\mathbb{C})$ given by $$F(X,Y,Z)=(X^2-Z^2)^2-Y^2(2YZ+3Z^2)$$ is a rational curve. I tried to do a quadratic transformation by determining $F(\frac{1}{X},\frac{1}{...
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1answer
32 views

Method of finding Arc length parameterization of a 3d curve

r(t) = cos^3 t i + sin^3 t j; 0 < t < pi/2. r'(t) = -3cos^2 t sin t + 3 sin^2 t cos t ||r'(t)|| = 3 sin t cos t Now to find the arc length parameterization, we need S = integration from t0 ...
0
votes
0answers
47 views

Rational maps between affine varieties

If I want to check that the map $$\phi:C_1\rightarrow C_1,\hspace{0.5cm}\phi(x,y)=(\phi_1(x,y),\phi_2(x,y))$$ between two affine plane curves is rational I just should check that $\phi_1$ and $\...
3
votes
1answer
56 views

Rationality of the Lemniscate.

This question is exercise 2 of Chapter 4 in Kunz' textbook of algebraic curves. Let $f$ be the lemniscate with equation $$(X^2 + Y^2 )^2 = α(X^2 − Y^2) \;\; (\alpha \in K^\times )$$ and let ...
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vote
1answer
85 views

Is every Riemann surface a 2-sheeted covering?

Given an algebraic curve $X$ over $\mathbb{C}$, i.e. a Riemann surface and a fixed set of pairs of points $S=\{(p_1,q_1),...,(p_1,q_1)\}$ is there an algebraic curve Y, possibly singular, and a map $f:...
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votes
3answers
59 views

Looking for the equation or algorithm for a mystery dataset [closed]

I'm a programmer by trade, although I did both A-level and engineering maths at University, I'm a little rusty. I'm trying to reverse engineer a pretty shoddy bit of legacy code. I have two sets of ...
4
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1answer
35 views

Gluing together Riemann surfaces, don't see why $Z$ is union of two compact sets.

Consider Lemma 1.7 from page 60-61 of Miranda's Algebraic Curves and Riemann Surfaces. For a link to the book, see here. Lemma 1.7. With the above construction, $Z$ is a compact surface of genus $...
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1answer
75 views

Cohomology and normalization of a curve

Suppose $C$ is a projective curve such that $H^1(C,\mathcal{O}_C)=0$ and let $\pi:\tilde{C}\longrightarrow C$ be the normalization morphism. How can I show that $H^1(\tilde{C},\pi^\ast\mathcal{O}_C)=0$...
3
votes
1answer
59 views

Intersection of following pair of parabolas at infinity? [closed]

What is the intersection multiplicity of the following pair of parabolas at infinity:$$y = x^2,\text{ }y = x^2 + 1?$$
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1answer
37 views

Maps induced by divisors and degree

I feel very stupid to ask this question but I've serached everywhere and I've not found a clear (to me) answer. A well known result in Curve theory (over $k=\bar{k}$) states that a divisor $D$ on a ...
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2answers
59 views

The Hirzebruch surface $\mathbb F_n$ contains a unique irreducible curve with negative self-intersection.

Consider the $n$-th Hirzebruch surface $\mathbb F_n:=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}\otimes\mathcal O_{\mathbb P^1} (n))$, and let $C_0\subseteq \mathbb F_n$ a section of the ruling ...
0
votes
0answers
7 views

C: a smooth projective curve/k. $x\in\bar{k}(C).dx$ is a basis for $\Omega_C,\Rightarrow\bar{k}(C)$ is a finite separable extension of $\bar{k}(x)$.

Let C be a smooth projective curve over a field $k$ and $x\in\bar{k}(C)$. If $dx$ is a basis for $\Omega_C$, then $\bar{k}(C)$ is a finite separable extension of $\bar{k}(x)$.
1
vote
1answer
55 views

$\mathbb P^1\times \mathbb P^1$ is minimal

I'm trying to prove that the complex surface $\mathbb P^1\times \mathbb P^1$ is minimal. I'd like to prove it directly, namely by showing that there are not exceptional curves. Suppose that $E$ is a ...
5
votes
1answer
71 views

Degree of $f:\mathbb{P}^1_k\rightarrow \mathbb{P}^1_k$

Let $k$ be an algebrically closed field and consider $\mathbb{P}^1_k$ the $1$-dimensional projective space over $k$. My question is the following: Let consider $f:\mathbb{P}^1_k\rightarrow \...
2
votes
1answer
67 views

Geometrically ruled surface, sections and intersection numbers.

Consider a geometrically ruled surface $\pi: S\longrightarrow C$ where both $S$ and $C$ are projective, non-singular and complex ($C$ is obviously a curve). By using Tsen theorem one can show that ...
3
votes
0answers
51 views

Tricks to find $\dim H^0(O(D)(n))$

Let $X$ be a curve of genus $g\neq 0$ and $D$ a divisor on $X$. If $n \in \mathbb{Z}$ and $O(D)(n)=O(D)\otimes O(n)$ is the twist of $O(D)$ by $n$, there is a manageable proceedings to find the ...