The multiplicity of cyclic coverings of a singularity of an algebraic variety

Let V be an affine algebraic variety, and let p∈ V be a singular point. For a regular function g on V such that $g(p)=0$ and for a positive integer n, we consider the cyclic covering φₙ\: Vₙ → V of degree n branched along the hypersurface defined by g. We will prove that for sufficiently large n, the tangent cone of Vₙ at φₙ⁻¹(p) is, as an affine variety, the product of the tangent cone of the branch locus and the affine line. In particular, the multiplicity of the singularity φₙ⁻¹(p) ∈ Vₙ, which is a function of n determined by V and g, remains constant for sufficiently large n. This result generalizes Tomaru's theorem for normal surface singularities.