Ordinaries

Ordinaries is a term used in singularity theory and algebraic geometry to describe particularly simple or “ordinary” singularities of geometric objects, such as curves, surfaces, or more general varieties. The idea is to single out singularities that, locally, behave as simply as possible given their multiplicity, often consisting of several smooth branches crossing transversely.

A common precise notion is that of an ordinary multiple point. Let p be a point on a hypersurface defined by f = 0 in a neighborhood over an algebraically closed field. The multiplicity m of p is the smallest degree at which the homogeneous component f_m of the Taylor expansion at p is nonzero. The point p is called ordinary if f_m factors as a product of m distinct linear forms; equivalently, the tangent cone f_m = 0 is reduced and consists of m distinct tangent directions. When m = 1, the point is smooth. When m = 2 and the tangent cone factors into two distinct lines, the point is an ordinary double point, or node, which locally looks like two smooth branches crossing transversely.

In practice, ordinaries are used as a baseline for classifying singularities and guiding resolution strategies. They contrast with more complicated singularities, such as cusps or higher-order tangencies, where branches may have non-transverse intersections or higher-order contact.

In two dimensions, familiar examples include a node | y^2 − x^2 = 0, which locally splits into two crossing lines, and a cusp | y^2 − x^3 = 0, which is not ordinary. The concept generalizes to higher dimensions, maintaining the core idea of transverse, distinct tangent directions at the singular point.