Isomorphism of affine varieties

Question

Let $X\subseteq\mathbb{C}^n$ and $Y\subseteq\mathbb{C}^m$ be two complex affine varieties (zero sets of systems of polynomials). Suppose that $$f:X\to Y$$ is a polynomial map, or more precisely, it is given explicitly by $$f(x)=(f_1(x),\ldots,f_m(x)),$$ where $f_i:\mathbb{C}^n\to\mathbb{C}$ are polynomials in $n$-variables.

Question: If $f$ is a bijection, is it necessarily an isomorphism of complex affine varieties?

Answer 1

The answer is no. A standard counter example is the following: let $X = \mathbb{C}$ and $Y = V(y^2-x^3) \subset \mathbb{C}^2.$ Then the map $f : X \rightarrow Y$ given by $t \mapsto (t^2,t^3)$ is a bijection, but not an isomorphism.

One way to see that it couldn't be an isomorphism is to note that $X$ is smooth while $(0,0)$ is a singular point of $Y$.