When is the preimage of a hypersurface a hypersurface

Question

Let $f:X \to Y$ be a proper, dominant morphism of projective varieties and $H_Y$ a hypersurface in $Y$. Is the fiber $f^{-1}(H_Y)$ a hypersurface in $X$?

Answer 1

Yes, given a proper dominant morphism $f:X\to Y$ of irreducible varieties (projective or not!), the inverse image of a hypersurface $H\subset Y$ is a hypersurface $f^{-1}(H)\subset X$.
Note that $f$ is surjective because $f(X)$ is dense (by hypothesis) and closed (by properness).
We can now apply Hartshorne, Chapter II, Exercise 3.22(a), page 95, to conclude that the codimension of $f^{-1}(H)\subset X$ is $1$.
More precisely: we can assume that $H$ is irreducible and then every irreducible component of $f^{-1}(H)$ which has image $H$ is of codimension $1$.

Answer 2

It is, if your definition of a variety includes irreducibility (otherwise, such a preimage could contain a component of $X$).