Suppose that $X$ and $Y$ are complex algebraic varieties with $Y$ affine. Suppose that $f:X\rightarrow Y$ is a variety morphism. I am interested in finding conditions on the $\mathbb{C}$-algebra morphism $f^*:\mathbb{C}[Y]\rightarrow\mathbb{C}[X]$ under which the image of $f$ is open. I would appreciate any and all references/suggestions.

Thanks!