Multiplicity Along Points of a Radicial Covering of a Regular Variety

Abstract

We study the maximal multiplicity locus of a variety X over a field of characteristic $\mathit{p}>0$ that is provided with a finite surjective radicial morphism $\mathit{\delta }:\mathit{X}\to \mathit{V}$, where V is regular, for example, when $\mathit{X}\subset {\mathbb{A}}^{\mathit{n}+1}$ is a hypersurface defined by an equation of the form ${\mathit{T}}^{\mathit{q}}-\mathit{f}({\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}})=0$ and δ is the projection onto $\mathit{V}:=\mathrm{Spec}(\mathit{k}[{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}}])$. The multiplicity along points of X is bounded by the degree, say d, of the field extension $\mathit{K}(\mathit{V})\subset \mathit{K}(\mathit{X})$. We denote by ${\mathit{F}}_{\mathit{d}}(\mathit{X})\subset \mathit{X}$ the set of points of multiplicity d. Our guiding line is the search for invariants of singularities $\mathit{x}\in {\mathit{F}}_{\mathit{d}}(\mathit{X})$ with a good behavior property under blowups ${\mathit{X}}^{\prime }\to \mathit{X}$ along regular centers included in ${\mathit{F}}_{\mathit{d}}(\mathit{X})$, which we call invariants with the pointwise inequality property.

A finite radicial morphism $\mathit{\delta }:\mathit{X}\to \mathit{V}$ as above will be expressed in terms of an ${\mathcal{O}}_{\mathit{V}}^{\mathit{q}}$-submodule $\mathcal{M}\subseteq {\mathcal{O}}_{\mathit{V}}$. A blowup ${\mathit{X}}^{\prime }\to \mathit{X}$ along a regular equimultiple center included in ${\mathit{F}}_{\mathit{d}}(\mathit{X})$ induces a blowup ${\mathit{V}}^{\prime }\to \mathit{V}$ along a regular center and a finite morphism ${\mathit{\delta }}^{\prime }:{\mathit{X}}^{\prime }\to {\mathit{V}}^{\prime }$. A notion of transform of the ${\mathcal{O}}_{\mathit{V}}^{\mathit{q}}$-module $\mathcal{M}\subset {\mathcal{O}}_{\mathit{V}}$ to an ${\mathcal{O}}_{{\mathit{V}}^{\prime }}^{\mathit{q}}$-module ${\mathcal{M}}^{\prime }\subset {\mathcal{O}}_{{\mathit{V}}^{\prime }}$ will be defined in such a way that ${\mathit{\delta }}^{\prime }:{\mathit{X}}^{\prime }\to {\mathit{V}}^{\prime }$ is the radicial morphism defined by ${\mathcal{M}}^{\prime }$. Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on ${\mathcal{O}}_{\mathit{V}}^{\mathit{q}}$-submodules $\mathcal{M}\subset {\mathcal{O}}_{\mathit{V}}$.