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Let $Y=\{(a,b,c) \in \mathbb{A}^{3}: a^{2}-a^{2}b^{2}+c^{3}=0\}$. How can we parametrize $Y$ so that we can find the irreducible components of $Y$?

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up vote 1 down vote accepted

I'll change the notation and say we are studying $Y=V(x^2-x^2y^2+z^3)\subset \mathbb A^3_k$ .

The polynomial $f(x,y,z)=x^2-x^2y^2+z^3=z^3+x^2(1-y)(1+y)$ is irreducible in $k[x,y,z]=k[x,y][z]$ by Eisenstein's criterion applied to the prime $1-y\in k[x,y]$

Hence, for any field $k$, the closed subset $Y\subset \mathbb A^3_k$ is irreducible .

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