Purity of branch, critical and discriminant locus

Abstract

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.