Let $h:X\to Y$ a morphism of varieties. It is generally true that $h(X)$ is a subvariety of $Y$?

Question

I am a little confused with this, my intuition tells me that this is not true but I don't know what counterexample I should take, could someone help me? Thank you.

Answer 1

You might find this wikipedia article useful: https://en.wikipedia.org/wiki/Constructible_set_(topology)

It gives a counterexample to your question $A^2 \to A^2$ given by $(x,y) \mapsto (x,xy)$ but also explains an alternative to your question.