What can we say about the image of a regular map?

Question

If we have an irreducible subvariety of complex affine space $A_{\mathbb{C}}$, is the image under a regular (i.e., given by polynomials) map also irreducible? is it an irreducible subvariety of the target space?

Answer 1

Just to have this answered:

Yes, the image of an irreducible subvariety of $\mathbb{A}^n$ is irreducible. In fact, the image of any irreducible space under a continuous map is irreducible. This is just an exercise in point-set theory. It is proved in a similar way to the fact that the image of a connected set is connected.

The image of a variety need not be a subvariety though (by this I assume you mean a closed algebraic subset), as Cantlog points out in the comments. The most common example is the morphism $\mathbb{A}^2\to\mathbb{A}^2$ given by $(x,y)\mapsto (x,xy)$. The image of $\mathbb{A}^2$ is not closed. It is "constructible" though, as Cantlog points out. It turns out that the image of any variety under a morphism is constructible (this falls under the more general Chevalley's Theorem).