Singularities of ramification locus

Question

Let $f: X \to Y$ be a finite morphism of smooth projective varieties over $\mathbb{C}$. Let $R \subseteq X$ be the subvariety of all points where $f$ is ramified. Under what assumptions can we say that $R$ is smooth? Can you give me easy counter examples where it is not?

Answer 1

Counterexample where $R$ is not smooth: $$f:\mathbb P^2(\mathbb C)\to \mathbb P^2(\mathbb C):(x:y:z)\mapsto (x^2:y^2:z^2) $$ The ramification locus is the singular variety $R=V(xyz)\subset \mathbb P^2(\mathbb C)$, the union of three lines.