Dimension and morphism with finite fibers

Question

I'm studying the dimension of projective varieties and in the literature I'm reading I have the following statement:

"If $f : X → Y$ is a morphism with finite fibers, i. e. such that $f^{−1}(P)$ is a finite set for all P ∈ Y , we would expect that dim $ X ≤ \text{dim }Y$ .

Can someone provide some details regarding this "expectation"? It is clear to me that we would expect a lower dimension if $f$ is injective, since we then can perceive $X$ as "contained in" $Y$.

Answer 1

The "expectation" is actually a theorem:

Let $f:X\to Y$ be a morphism of varieties of finite type over a field $k$.
If the fibers of $f$ are finite sets, then $\operatorname {dim} X\leq \operatorname {dim} Y$

Remarks
a) Notice that projectivity of either variety is irrelevant and that $k$ is not assumed algebraically closed.
b) Morphisms of finite type with finite fibers are called quasi-finite.
c) A quasi -finite morphism is finite iff it is proper.
d) A non-trivial open immersion is the prototype of a quasi-finite but not finite morphism .
e) More generally, any quasi-finite morphism is obtained by deleting a closed subset upstairs from a finite morphism: this is a consequence of the difficult Zariski Main Theorem.