Ordinary isolated singularities of algebraic varieties (Q2182748)

Let \(P\) be an isolated singularity of an algebraic variety \(V\). The target of the paper is a natural extension of the notion of ordinary singularities of curves to varieties of arbitrary dimension. The authors consider the case where the normalization of \(V\) at \(P\) is regular around the point, a situation for which they provide huge classes of examples. In this case, the authors say that \(P\) is an ordinary singularity of \(V\) when the projectivized tangent cone at \(P\) is reduced. For ordinary singularities, the authors prove that the projectivized tangent cone is the union of \(e\) linear spaces of the same dimension, \(e\) being the multiplicity of \(P\). When the linear spaces that compose the projectivized tangent cone are in general position, the authors prove that also the affine tangent cone at \(P\) turns out to be reduced.