Multiplicity along points of a radicial covering of a regular variety (Q2113442)

The purpose of this paper is the study of the maximum multiplicity locus of a variety \(X\) defined over a field of positive characteristic \(p>0\), in the case in which \(X\) comes together with a finite surjective radicial morphism \(\delta: X\to V\) with \(V\) regular. If the generic rank of the morphism is \(d\), then the maximum multiplicity of \(X\) is bounded by \(d\). The set of points of multiplicity \(d\) is denoted by \(F_d(X)\), which is assumed to be non-empty, and suppose that \(d=p^e\), for some \(e\in {\mathbb Z}_{>0}\). The goal is to find invariants of singularities for the points in \(F_d(X)\) that do not get worse after blow ups at regular centers contained in \(F_d(X)\). Such invariants will be called \textit{invariants with the pointwise inequality property}. The approach here will be to pay attention to \(\delta(F_d(X))\) which is homomorphic to \(F_d(X)\) and define the invariant of \(x\in F_d(X)\) really for \(\delta(x)\). This is in principle a plausible approach because it also can be proved that \(Y\subset F_d(X)\) is an integral regular subscheme if an only if its schematic image \(\delta(Y)\subset V\) is regular, and then there is a commutative diagram of blow ups and finite surjective morphisms, \[ \begin{tikzcd} X \arrow[r] \arrow[d,"\delta"] & X_1 \arrow[d, "\delta_1"] \\ V \arrow[r] & V_1 \end{tikzcd}\tag{1} \] From he choice of the center, the maximum multiplicity of \(X_1\) is less than or equal to \(d\). As an example, suppose that \(V=\mathrm{Spec}(S)\) with \(S\) of finite type over a field \(k\), that \(X\subset V\times {\mathbb A}_k^1\) is defined by some polynomial \[ p(T)=T^d+f_1T^{d-1}+\ldots+f_d\in S[T], \] and if \(d\) is not divisible by the characteristic of \(k\). Then it is possible to define an ideal \(J\subset S\) and a positive integer \(b\) so that \[\delta(F_d)=\mathrm{Sing}(J,b):=\{\xi \in V: \nu_{\xi}(J)\geq b\},\] where \(\nu_{\xi}\) denotes the order at the regular local ring \({\mathcal O}_{V,\xi}\). And this description is valid after blowing up ar equimultiple regular centers using commutative diagrams as in (1). The ideal \(J\) is obtained from applying certain homogeneous polynomials to the coefficients of \(p(T)\) that, among other properties, are invariant under translations of the form \(T\to T+s\), with \(s\in S\). The case treated in this paper includes situations in which \[ p(T)=T^q-f(x_1,\ldots, x_n) \] where \(q\) is a power of the characteristic \(p>0\) of \(k\). Here, translations of the form \(T'=T+s\) lead to \(T^{'q}-(f+s)\), hence the ideal \(\langle f\rangle\) is not a good substitute for \(J\). Instead, it makes sense to consider \(f\) up to equivalence: \(f\sim f'\) if \(f-f'\in S^q\). This leads to considering the \(S^q\)-submodule \(M\subset S\) generated by \(f\) and to considering \(S^q\)-submodules of \(S\) up to the equivalence, \(M\sim M'\), if \(M+S^q=M'+S^q\). In addition, the embedding \(X\subset {\mathbb A}^{n+1}_k\) is not necessary, and \(X\) can be determined as a \(V\)-scheme by considering the \(S^q\)-subalgebra of \(S\) generated by \(f\): \(S^q[f]\). After the blow up at a regular equimultiple center, a notion of transform of the \(S^q\)-module \(M\) is defined so that given a commutative diagram as (1), the transform of \(M\) is, up to equivalence, the module \(M_1\) associated to \(X_1\). Invariants for the singularities of maximum multiplicity of \(X\) are defined in terms of \(M\). These invariants induce a stratification of the top multiplicity locus of \(X\).