Michigan Math. J. 71 (1), 47-104, (March 2022) DOI: 10.1307/mmj/20195775
D. Sulca, O. E. Villamayor U.
We study the maximal multiplicity locus of a variety X over a field of characteristic that is provided with a finite surjective radicial morphism , where V is regular, for example, when is a hypersurface defined by an equation of the form and δ is the projection onto . The multiplicity along points of X is bounded by the degree, say d, of the field extension . We denote by the set of points of multiplicity d. Our guiding line is the search for invariants of singularities with a good behavior property under blowups along regular centers included in , which we call invariants with the pointwise inequality property.
A finite radicial morphism as above will be expressed in terms of an -submodule . A blowup along a regular equimultiple center included in induces a blowup along a regular center and a finite morphism . A notion of transform of the -module to an -module will be defined in such a way that is the radicial morphism defined by . 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 -submodules .