  To a dominant morphism $\pi:X/S \to Y/S$ of N{\oe}therian integral
$S$-schemes one has the inclusion $C_\pi \subset B_\pi$ of the critical locus
in the branch locus of $B_\pi$. Conditions on the relative differentials
$\Omega_{X/Y}$, $\Omega_{X/S}$, and $\Omega_{Y/S}$ are stated that imply bounds
on the codimensions of $ C_\pi$ and $ B_\pi$. We also give bounds on the
codimension of the discriminant locus, proving and generalising a conjecture of
I. Dolgachev.
