Ordinary isolated singularities of algebraic varieties

Let A be the local ring,at a singular point P, of an algebraic variety V ⊂ Ar+1 of multi-k tiplicity e = e(A) > 1. If V is a curve in [13] P was said to be an ordinary singularity when V has e (simple) tangents at P or equivalently when the projectivized tangent cone Proj(G(A)) of V at P is reduced (in which case consists of e points). In this paper we show that the definition of ordinary singularity has a natural extension to higher dimen- sional varieties, in the case in which P is an isolated singularity and the normalization A of A is regular. In fact we define P to be an ordinary singularity if the projectivized tangent r cone Proj(G(A)) of V at P is reduced i.e. is a variety in Pk. We prove that an ordinary singularity has multilinear projectivized tangent cone that is a union of e linear varieties L1,...,Le. In the case in which L1,...,Le are in generic position we show that the affine tangent cone is also reduced and then multilinear . Finally we show how to construct wide classes of parametric varieties with regular normalization at a singular isolated ordinary point.