Ordinary Isolated Singularities of Algebraic Varieties

Abstract

Let A be the local ring, at a singular point P, of an algebraic variety $V\subset {\mathbb {A}}^{r+1}_k$ of multiplicity $e=e(A)>1$. If V is a curve in Orecchia (Can Math Bull 24:423–431, 1981) P was said to be an ordinary singularity when V has e (simple) tangents at P or equivalently when the projectivized tangent cone $\mathrm{{Proj}}(G(A))$ 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 dimensional varieties, in the case in which P is an isolated singularity and the normalization $\overline{A}$ of A is regular. In fact, we define P to be an ordinary singularity if the projectivized tangent cone $\mathrm{{Proj}}(G(A))$ of V at P is reduced, i.e. is a variety in ${\mathbb {P}}^{r}_k$. We prove that an ordinary singularity has multilinear projectivized tangent cone that is a union of e linear varieties $L_1,\ldots ,L_e$. In the case in which $L_1,\ldots ,L_e$ 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.