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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61 views

Smooth affine plane curve with non-trivial cotangent sheaf?

Question: Let $A = \mathbb C[x,y]/(f)$ be a non-singular plane curve. Under what conditions is the module of Kahler differentials $\Omega_A^1$ (over $\mathbb C$) a free module? I am not sure what ...
3
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2answers
59 views

Proving that set of points $(x,y)$ in $\mathbb{R}^2$ satisfying $y - \cos(x)= 0$

How can one prove that set of points $(x,y)$ in $\mathbb{R}^2$ satisfying $y-\cos(x)=0$ is not a algebraic curve. That is there does not exist a polynomial $f(x,y)$ in two variables $x$ and $y$ and ...
3
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1answer
51 views

If the $m-1$ first derivatives of a rational function vanish at a point, does the function have a zero of order $m$ at that point?

Let $C\subseteq\mathbb{P}^{2}$ be a projective smooth algebraic curve, and let $$ \alpha:K(C)\rightarrow K(C) $$ be a derivation, i.e. $\alpha$ is a $K$-linear map such that $$ \alpha(fg)=f\alpha(g)+...
2
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1answer
37 views

Computing the restriction $T_{\mathbb{P}^3|X}$ for twisted cubic in $\mathbb{P}^3$

Let $i:X=\mathbb{P}^1\to\mathbb{P}^3$ be a twisted cubic given by the embedding $(u:v)\mapsto(u^3: u^2v: uv^2: v^3)$. How to compute $T_{\mathbb{P}^3|X}$?
3
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1answer
49 views

Are all rationally parametrized plane curves algebraic? How does one find their degree?

Suppose a plane curve is given parametrically by $x=p(t),y=q(t)$, where $p,q$ are rational functions. I originally assumed that this means that the parametrized curve is algebraic, i.e. that it is the ...
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vote
1answer
30 views

Schaum's Differential Geometry exercise on curvature

Page 72 exercise 4.5, there is the following situation: There is a curve $\underline{x}(t)$ with $t$ not a natural parameter. I have to find the curvature vector $\underline{k}$ and the curvature $k$ ...
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2answers
77 views

Generator for Kahler differentials of an affine elliptic curve

Consider the affine (nonsingular) elliptic curve $A = \mathbb C[x,y]/(y^2-x^3+x)$. Since the cotangent bundle is trivial, $\Omega_A^1 = A\,dx\oplus A\,dy /(2y\,dy - (3x^2-1)\,dx)$ is a free $A$-...
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2answers
72 views

Two polynomials $f,g \in K[x,y]$ ring. Prove that $K[x,y]/(f,g)$ is finite dimensional vector space

Let $f,g \in K[x,y]$ be polynomials with no common factor. Prove that $K[x,y]/(f,g)$ is a finite dimensional vector space. I know there are non-zero (this word is correct?) $r(x)$ and $s(x)$ in the ...
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votes
2answers
23 views

Area using definite integrals with a straight line

I'm really stuck on this. Say you have a curve $y = 3x - x^2$ that cuts the x-axis at points $O$ and $A$, and meets the line $y = -3x$ at the point $B$. How would you find the area of this shaded ...
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50 views

The normal bundle of conic

Let $C\subset\mathbb{P}^2\subset\mathbb{P}^n$ be a smooth conic (everything is over the field $\mathbb{C}$). I want to compute $T_{\mathbb{P}^n|C}$ and $N_{C/\mathbb{P}^n}$. Let $z_0,z_1,...,z_n$ be ...
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0answers
42 views

The divisor of a nonconstant function on a smooth curve

Let $C/K$ be a smooth curve and $f \in K(C)$ be a function. Then by identifying $f$ with a rational map, we can get a 1-1 correspondence with maps $C \to \mathbf{P}^1$, with one direction being given ...
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votes
0answers
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
64 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
40 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
86 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
31 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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0answers
27 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 ...
4
votes
1answer
84 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
votes
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 ...
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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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votes
0answers
80 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
41 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 \...
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votes
2answers
165 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)+...
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79 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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0answers
73 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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votes
0answers
26 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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votes
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 ...
2
votes
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
votes
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
137 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
64 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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votes
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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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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0answers
31 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
46 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}...
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0answers
82 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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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
7 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 ...
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0answers
30 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}{...
0
votes
1answer
33 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 ...
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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 ...
1
vote
1answer
88 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:...
0
votes
3answers
62 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 ...