A point on an algebraic variety or scheme at which the local structure fails to be smooth or regular; equivalently, a point where the local ring is not regular or the Jacobian matrix of defining equations drops rank.

Principle

Singular points are characterized by a failure of the expected local dimension or regularity: the tangent space has dimension greater than the variety's local (Krull) dimension, which algebraically corresponds to generators of the maximal ideal requiring more relations than in the smooth case.

Demonstration

On a plane curve defined by f(x,y)=0, a point p is singular when the partial derivatives ∂f/∂x and ∂f/∂y both vanish at p. For example, f(x,y)=y^2-x^3 has the origin as a singular point because its gradient is zero there.

Misapplication

Calling any solution of the system of equations 'a singularity' even when the local Jacobian has full rank — for instance confusing an ordinary solution point with a singular point or misusing 'singular' to mean 'exceptional' in a nonlocal sense.

Consequence

At a true singular point one cannot apply the standard smooth tools: no single well-defined tangent line or manifold structure, tangent-space computations must account for embedded or excess components, and invariants (multiplicity, δ-invariant, Milnor number) change accordingly.

Reversal

A smooth (regular) point, where the local ring is regular and the Jacobian has maximal rank, yielding a well-defined tangent space of expected dimension.

Boundary

Applies scheme-theoretically to points of varieties or schemes over any base; excludes analytic or topological 'bad behavior' not witnessed algebraically. Isolated versus non-isolated singularities and scheme-theoretic embedded components are distinct subcases.

Semantic Tension

Overlaps with the differential-topology notion of a critical point (where a derivative vanishes) but differs because algebraic singularity uses scheme-theoretic regularity and multiplicity rather than smooth map criticality.

Synthesis

A singular point is where the algebraic structure locally ceases to be regular: algebraically indicated by a nonregular local ring or rank drop of the Jacobian, geometrically seen as a failure of a well-defined smooth tangent and often measurable by multiplicity-based invariants.