Finite morphisms and simultaneous reduction of the multiplicity

Let X be a singular algebraic variety defined over a perfect field k, with quotient field K(X). Let s≥2 be the highest multiplicity of X and let Fs(X) be the set of points of multiplicity s. If Y⊂Fs(X) is a regular center and X←X1 is the blow up at Y, then the highest multiplicity of X1 is less than or equal to s. A sequence of blow ups at regular centers Yi⊂Fs(Xi), say X←X1←⋯←Xn, is said to be a simplification of the multiplicity if the maximum multiplicity of Xn is strictly lower than that of X, that is, if Fs(Xn) is empty. In characteristic zero there is an algorithm which assigns to each X a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties β:X′→X of generic rank r≥1 (i.e., [K(X′):K(X)]=r). We will see that, when imposing suitable conditions on β, there is a strong link between the strata of maximum multiplicity of X and X′, say Fs(X) and Frs(X′) respectively. In such case, we will say that the morphism is strongly transversal. When β:X′→X is strongly transversal one can obtain information about the simplification of the multiplicity of X from that of X′ and vice versa. Finally, we will see that given a singular variety X and a finite field extension L of K(X) of rank r≥1, one can construct (at least locally, in étale topology) a strongly transversal morphism β:X′→X, where X′ has quotient field L.