##  [Singular Point](/singular-point-0) 

 Definition

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

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

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

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

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

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

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

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

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.