Let $ X, Y $ be separated finite type schemes over an algebraically closed field $ k $. Assume that $ Y $ is reduced. Let $ \phi : X \rightarrow Y $ be a morphism of schemes. Suppose that $ \phi $ gives a bijection on $ k $-points and an injection on $ S$-points for all $k$-schemes $ S$. Prove or disprove that $ \phi $ is an isomorphism (or add some extra hypotheses to ensure that $ \phi $ is an isomorphism).
When $ k = \mathbb{C} $, $ X $ is normal, an $ Y $ is normal and irreducible, then I have a proof which uses the following crazy fact:
If $ X $ and $ Y $ are irreducible varieties over $\mathbb{C} $ and $ Y $ is normal, then a morphism $ \phi $ inducing a bijection on $ \mathbb{C} $-points is an isomorphism.
My original question follows from this fact via a small tweaking of the usual Yoneda argument.
If anyone can give me a proof or reference for this last fact, I would be grateful too. I read it in the appendix of Kumar's book on Kac-Moody groups.
Edit: In light of some counterexamples, let me assume that X is irreducible and Y is normal and irreducible.
Best Answer
Trivial counterexample when $X$ is not connected: let $F$ be closed an nonempty in $Y$, $U:=Y\setminus F$ (assumed nonempty), and $X$ the disjoint sum of $U$ and $F$.
A bit less trivial with $X$ and $Y$ irreducible: $Y$= an irreducible curve with a node $y$, $X'$:=its normalization, $X$= $X'$ with one of the two points over $y$ removed.
The property holds, indeed, if $X$ is irreducible and reduced and $Y$ normal, assuming only that $X(\mathbb{C})\to Y(\mathbb{C})$ is bijective. In fact, in this case $f$ must be quasifinite and birational, hence an isomorphism if $Y$ is normal.