Purity of branch, critical and discriminant locus

Details Purity of branch, critical and discriminant locus Purity of branch, critical and discriminant locus

2006-09-11

arxiv.org/abs/math/0609300v3

Mathematics

Algebraic Geometry

This paper is withdrawn and will be superseded by new paper. The proofs are incomplete in this version

Scientific paper

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.

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