Examples of morphisms which are injective on complex points but not an isomorphism.

Question

I am trying to find some examples of:

A morphism $X →Y$ of complex affine schemes which is injective on complex points but is not an isomorphism onto a closed subscheme of $Y$.

Any ideals would be appreciate. If possible, I wish to know that how to approch this an where to get some ideas.

Thanks!

Answer 1

A simple source of examples is to take $X$ to be an open proper subscheme.

A variation on this that satisfies some other niceness properties one might have is to take $X$ to be the plane hyperbola $xy=1$ and $Y$ to be the $x$-axis, and the morphism to the the projection.