Possible dimension of branch locus in definition of finite cover of varieties/manifolds Given a finite morphism of (complex) varieties $f : X \longrightarrow Y$ ($\dim Y \ge 2$), is the branch locus $B \subset Y$ always a divisor in $Y$?
There seem to be differing accounts from different questions on this site (on branched covers and branch locus of $f: X \longrightarrow Y$ morphism) possibly depending on smoothness assumptions/whether the objects involved actually define manifolds when we're working over $\mathbb{C}$.
 A: Let me first point out a difference in terminology: different people use "branch locus" to refer to either the set in $X$ where $f$ is ramified (I usually call this the ramified locus) or the image of this under the map $f$ (this is what I usually consider the branch locus). Unfortunately, it seems like these terms are not standardized in the literature - the result I'll present below speaks of the branch locus in the first way, i.e. on $X$.

The big result in this area is called the "purity of the branch locus". This was originally proven by Zariski and Nagata in 1958 (On the purity of the branch locus of algebraic functions by Zariski, and Remarks on a paper of Zariski on the purity of the branch locus by Nagata). For better or worse, these papers were mostly written in the old style which I find a bit hard to read. A more modern expose is given in the Stacks Project as tag 0BMB:

Lemma. Let $f:X\to Y$ be a morphism of locally noetherian schemes. Let $x\in X$ and set $y=f(x)$. Assume

*

*$\mathcal{O}_{X,x}$ is normal,

*$\mathcal{O}_{Y,y}$ is regular,

*$f$ is quasi-finite at $x$,

*$\dim \mathcal{O}_{X,x} = \dim \mathcal{O}_{Y,y} \geq 1$,

*for specializations $x'\rightsquigarrow x$ with $\dim \mathcal{O}_{X,x'}=1$ our $f$ is unramified at $x'$.

Then $f$ is etale at $x$.

In your case, we can use this to arrive at the following, where I use "variety" to mean "scheme of finite type":
Lemma. Suppose $f:X\to Y$ is a map of varieties over $\Bbb C$. If $X$ is normal, $Y$ is regular, and $f$ is finite, then the branch locus is of pure codimension one (i.e. a divisor) inside $Y$.
To prove this, apply the lemma above, and then note that a finite surjective morphism preserves dimensions.
Removing either the assumption that $X$ is normal or $Y$ is regular should lead to counterexamples, but I don't have them immediately on hand for you. If you're interested in them, please leave a comment and I'll see what I can come up with.
