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I'm having trouble determining if the algebraic sets $Z(x^2-y^3)\subset \mathbb{A}^2$ and $Z(y^2-x^3-x^2)\subset\mathbb{A}^2$ are isomorphic over $\mathbb{C}$. My guess is that this boils down to determining if $\mathbb{C}[x,y]/(x^2-y^3)$ is isomorphic to $\mathbb{C}[x,y]/(y^2-x^3-x^2)$ but then again, I'm stuck.

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    Thanks for pointing it out!2012-10-24

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Thinking geometrically, we expect these varieties are not isomorphic, due to the fact that the first is a cusp, while the second is a node. One way to verify this is to consider the tangent cone of each. In the first case, we get $TC_{(0,0)}=V(x^2)$ which is interpreted as the line $x=0$ with multiplicity $2.$ In the second case, we get $TC_{(0,0)}=V(y^2-x^2)=V((y-x)(y+x))$ which is two distinct lines.

Since the tangent cone is an invariant under isomorphism, we see that there is no point on the first variety to correspond with the origin in the second, and vice-versa.

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    There are elementary proofs, but they are longer and quite silly compared to geometric arguments. This is really what algebraic geometry is about ...2012-10-24