Algebraic Curve

Algebraic Geometry
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Tsakanikas Nickos
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Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

Algebraic Curve

#1

Post by Tsakanikas Nickos »

Prove that the set \( \displaystyle \left\{ (t^2,t^3+1) \in \mathbb{C}^{2} \, \big| \, t \in \mathbb{C} \right\} \) defines an (affine) algebraic curve.
Papapetros Vaggelis
Community Team
Posts: 426
Joined: Mon Nov 09, 2015 1:52 pm

Re: Algebraic Curve

#2

Post by Papapetros Vaggelis »

If \(\displaystyle{\left(t^2,t^3+1\right)\in\mathbb{C}^2}\) is a typical element of the given set, then

by setting \(\displaystyle{x=t^2\,,y=t^3+1}\), we get :

\(\displaystyle{y-1=t^3\implies (y-1)^2=t^6=x^3\iff x^3-y^2+2\,y-1=0}\) .

On the other hand, if \(\displaystyle{f(x,y)=x^3-y^2+2\,y-1\in\mathbb{C}[x,y]}\), then

\(\displaystyle{f(x,y)=0\iff x^3-y^2+2\,y-1=0\iff x^3=(y-1)^2}\). So, by considering the family

\(\displaystyle{y-1=t\,x\,,t\in\mathbb{C}}\), we get :

\(\displaystyle{f(x,y)=0\,\land y-1=t\,x\iff x=t^2\,,\land y=t^3+1}\) and if \(\displaystyle{x=0}\)

then \(\displaystyle{y=1}\) so :

\(\displaystyle{\left\{\left(t^2,t^3+1\right)\in\mathbb{C}^2:t\in\mathbb{C}\right\}=V(f(x,y))}\) .
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