Find an infinite collection of maximal ideals containing $(x^2 - y^3) \subset \mathbb{C}[x,y]$

Question

What is an infinite collection of maximal ideals containing the ideal $I = (x^2 - y^3) \subset \mathbb{C}[x,y]$?

Answer 1

Hint: For every $\alpha \in \mathbb C$ consider the ideal $$ (x-\alpha^3,y-\alpha^2) \subset \mathbb C[x,y].$$ Can you show that every ideal of this form contains $I$?

Answer 2

Continuing from Hilbert's Nullstellensatz: suppose $(x^2-y^3)$ is contained in a maximal ideal $(x-a,x-b)$. Then we can write $x^2-y^3=p (x-a)+q (y-b)$ for polynomials $p,q$ and evaluating at $(a,b)$ gives that $a^2=b^3$. So your task is the same as showing the zero locus of $x^2-y^3$ is infinite. This is true since $\Bbb C$ is infinite and every pair $(z^3,z^2)$ lies in the zero locus of this polynomial.