Let $f : \mathbb{C}^3 \rightarrow \mathbb{C}^4$ be defined by $(s, t, u) \rightarrow (st, st^2+(1-s)u, st^3, 1-s)$, where $\mathbb{C}$ denotes the complex numbers.
Then for some irreducible polynomial $(x_1)^3 - x_3(1-x_4)^2$ in $C[x_1, x_2, x_3, x_4]$ the image of $f$ is properly contained in the affine variety $V((x_1)^3 - x_3 (1-x_4)^2)$
Then, I have to show that the image is dense in the variety. How to prove it?
Also, I think that $f(\mathbb{C}^3)$ is not open in $V((x_1)^3 - x_3(1-x_4)^2)$ either. Could anyone show me how to prove it rigorously?
