Multiplicity Along Points of a Radicial Covering of a Regular Variety

We study the maximal multiplicity locus of a variety X over field characteristic p>0 that is provided with finite surjective radicial morphism δ:X→V, where V regular, for example, when X⊂An+1 hypersurface defined by an equation form Tq−f(x1,…,xn)=0 and δ projection onto V:=Spec(k[x1,…,xn]). The along points bounded degree, say d, extension K(V)⊂K(X). denote Fd(X)⊂X set d. Our guiding line search invariants singularities x∈Fd(X) good behavior property under blowups X′→X regular centers included in Fd(X), which we call pointwise inequality property. A δ:X→V as above will be expressed terms OVq-submodule M⊆OV. blowup equimultiple center Fd(X) induces V′→V δ′:X′→V′. notion transform OVq-module M⊂OV to OV′q-module M′⊂OV′ such way δ′:X′→V′ M′. relies on techniques involving differential operators varieties also logarithmic operators. Indeed, different introduce stratification they define ideals obtained evaluating OVq-submodules M⊂OV.