March 2022 Multiplicity Along Points of a Radicial Covering of a Regular Variety
D. Sulca, O. E. Villamayor U.
Michigan Math. J. 71(1): 47-104 (March 2022). DOI: 10.1307/mmj/20195775


We study the maximal multiplicity locus of a variety X over a field of characteristic p>0 that is provided with a finite surjective radicial morphism δ:XV, where V is regular, for example, when XAn+1 is a hypersurface defined by an equation of the form Tqf(x1,,xn)=0 and δ is the projection onto V:=Spec(k[x1,,xn]). The multiplicity along points of X is bounded by the degree, say d, of the field extension K(V)K(X). We denote by Fd(X)X the set of points of multiplicity d. Our guiding line is the search for invariants of singularities xFd(X) with a good behavior property under blowups XX along regular centers included in Fd(X), which we call invariants with the pointwise inequality property.

A finite radicial morphism δ:XV as above will be expressed in terms of an OVq-submodule MOV. A blowup XX along a regular equimultiple center included in Fd(X) induces a blowup VV along a regular center and a finite morphism δ:XV. A notion of transform of the OVq-module MOV to an OVq-module MOV will be defined in such a way that δ:XV is the radicial morphism defined by M. 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 OVq-submodules MOV.


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D. Sulca. O. E. Villamayor U.. "Multiplicity Along Points of a Radicial Covering of a Regular Variety." Michigan Math. J. 71 (1) 47 - 104, March 2022.


Received: 25 July 2019; Revised: 18 January 2021; Published: March 2022
First available in Project Euclid: 4 May 2021

MathSciNet: MR4389671
zbMATH: 1505.14034
Digital Object Identifier: 10.1307/mmj/20195775

Primary: 14E15

Rights: Copyright © 2022 The University of Michigan


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Vol.71 • No. 1 • March 2022
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