Purity of branch, critical and discriminant locus

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

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