A point x on a scheme X whose local ring O_{X,x} is not a regular local ring; equivalently the embedding dimension (dimension of m_x/m_x^2) strictly exceeds the Krull dimension of O_{X,x}, indicating a local singularity in the sense of commutative algebra.

Principle

Regularity of a local ring is the organizing rule: a local ring is regular exactly when its maximal ideal can be generated by dim O_{X,x} elements. Failure of this equality signals excess directions in the Zariski tangent space and hence a singular point.

Demonstration

Concrete instance: the cusp defined by the affine curve Spec k[t^2,t^3] at the origin. Its local ring has embedding dimension 1 but requires an extra integral relation so the tangent (m/m^2) dimension exceeds the geometric dimension; the origin is therefore a nonregular point. More generally, plane curve singularities like y^2 = x^3 give the same phenomenon.

Misapplication

Mistakenly labeling every point with multiple incident branches as 'nonregular' without checking the local embedding dimension; e.g., a node and a cusp are both singular but their local algebraic properties differ and must be tested via the local ring, not only by counting branches.

Consequence

Presence of nonregular points prevents the scheme from being smooth; it enlarges tangent spaces, complicates deformation theory, and obstructs the direct application of tools that require regularity (e.g., certain vanishing theorems, simple descriptions of dualizing complexes).

Reversal

A regular point is one where O_{X,x} is a regular local ring: the embedding dimension equals the Krull dimension, tangent space has the minimal possible dimension, and locally the scheme is nonsingular (smooth over a field when combined with separability conditions).

Boundary

This notion applies pointwise to Noetherian schemes (or locally Noetherian) via their local rings. It does not by itself capture nonreduced pathologies or failures of normality; a point can be nonregular while the scheme remains reduced or even normal in some cases. Detection typically requires local algebra computations (depth, embedding dimension) and may be subtle in non-Noetherian contexts.

Semantic Tension

Close concepts include 'singular point', 'nonnormal point', and 'nonreduced point'. 'Nonregular' is an algebraic condition on the local ring (embedding vs Krull dimension), while 'singular' is often used geometrically for failure of smoothness; the two overlap but emphasize different technical checks.

Synthesis

A Nonregular Point is the local algebraic signature of a singularity: it is a point where the local ring fails the regularity equality between generators of the maximal ideal and ring dimension, producing an enlarged Zariski tangent space and the usual downstream obstructions in geometry and deformation theory.