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Questions tagged [algebraic-curves]

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 of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

If $K$ is a field, then an algebraic curve is an equation $f(X,Y)=0$ where $f(X,Y)$ is a polynomial with coefficients in the field. In other words, it is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

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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 having integer coefficients such that $f(x,y) =…
etet112
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Proving the existence of a particular linear subvariety

I'm trying to prove that if $V$ is a non-empty linear subvariety then there is an affine change of coordinates $T$ of $ \Bbb A^n $ such that $V^T = V(X_{m+1}, \ldots, X_n) $. A set V in $ \Bbb A^n(k) $ is called a linear subvariety of $ \Bbb A^n(k)…
Ghassen Hamrouni
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Singular Points on Irreducible Cubic Curves Defined over Not Necessarily Algebraically Closed Fields

Let $C$ be a cubic curve defined over a field $k$. Take, for example, an affine curve: $$ C = \{(x,y) \in k\times k : a x^{3} + b x^{2} y + c x y^{2} + d y^{3} + e x^{2} + f x y + g y^{2} + h x + i y + j = 0 \} $$ I know that if the curve were…
user115624
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Help in this very basic example in algebraic curves

I'm trying to understand this example: I didn't understand why the second factor describes a point of intersection $q$, since the second factor doesn't vanish at $q$. Anyone can clarifies this for me please? Thanks
user75086
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Intersection number of the tangent at the Inflexion point of $y=x^3$

We know that the intersection number of this curve $f=y-x^3$ and its tangent at the origin is $3$. I'm trying to use this method described in the Fulton's book: Following this definition we have the intersection number equals to $1$, because $m=1$…
user42912
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Nonsingular affine curve which is not unmixed

Let $C$ be any nonsingular curve in $A^3_{\mathbb C}$. Can a point be an irreducible component of $C$? I am not able to find an example of such $C$.
A.G
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Number of intersection multiplicity points .

I need help for the following problem : Consider $C_1 = V(F_1)$ and $C_2=V(F_2)$ be algebraic curves in $\mathbb P (\bar K )$ (where $K$ is a field,) without a common component and $F_1, F_2 \in \bar K [X,Y,Z]$ are homogenous with $\deg(F_1 ) \le…
Theorem
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Large Intersection Multiplicity

A cubic curve, say, $x^3+y^3=1$ and some quadratic curve $f(x,y)=0$ generally have six intersection points in $\mathbb{CP}^2$. Question: If all the intersection points coincide, what will be the exact form of $f(x,y)$?
zy_
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Polar plane curves algebraic

Let $h\in \mathbb{R}[x,y]$ be a nonzero polynomial and define a plane curve in polar coordinates as $r(\theta) = h(\cos\theta,\sin\theta)$. For all the examples I've looked at, it seems like we can describe this curve as a zero set of a polynomial…
asdf
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Rational Parametrization of cubic curve $x^3+y^3=1$

How to rationally parametrize the cubic curve $x^3+y^3=1$? It's a slice of Fermat cubic, and the Fermat cubic does have rational parametrization, so I think it should also have rational parametrization. I tried $u=x+y, v=x-y$ and got…
MaudPieTheRocktorate
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Apply Riemann-Roch to describe the two-torsions points on Jacobian of genus 2 hyperelliptic curve

I am reading the following paper: https://arxiv.org/abs/1101.4792. In lemma 4, they prove: If $P_1,\ldots,P_6$ are the Weierstrass points of a genus 2 hyperelliptic curve $H$, then every element of the $2$-torsion subgroup $J[2]$ of the Jacobian…
mainomai
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intersection multiplicity from kirwan’s book

I'm working on F. Kirwan's Complex Algebraic Curves. To define intersection multiplicity, Kirwan choose some special projective coordinates and calculate the resultant. She claims before the definition that "In order to show that the definition is…
Ryan
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Why are all non-singular curves absolutely irreducible?

I was reading Judy Walker's book Codes and Curves, and one of the exercise's in the book (ex. 4.6) was proving that every non-singular curve is abaolutely irreducible. I'm not so familiar with algebric geometry, and I'm not so sure how to approach…
Yotam D
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Find the singular point

Let $f(x, y)=(x-y)^2$. We want to find the singular points. We do the following: Let $P=(a, b)$ be the singular point. $$f(a,b)=0 \Rightarrow (a-b)^2=0 \Rightarrow a=b \\ \frac{\partial{f}}{\partial{x}}(a,b)=0 \Rightarrow 2(a-b)=0 \Rightarrow a=b…
user175343
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Intersection Multiplicites

I have the following problem; Let $C = \{Q:=x_0x_2^2 -x_1(x_1-x_0)(x_1+x_0)=0\}$ and $L = \{ax_0 + bx_1 = 0\}$ be two projective curves with $(a,b) \ne (0,0)$. Let $p=[0,0,1]$, then I am asked to calculate the intersection multiplicity $I(p, C,…
Tim
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