##  [Double Point](/double-point-0) 

 Definition

A singular point on a curve or surface where two smooth local branches meet transversely, producing a simple crossing of multiplicity two; locally modeled by the equation xy = 0 in appropriate coordinates.

 

 

 

 

 

 





## Principle

Principle

Transverse intersection of two smooth local components with intersection multiplicity two: the tangent directions are distinct and the local analytic type decomposes into two smooth branches crossing simply.

 

 

 

 

 





## Demonstration

Demonstration

A planar algebraic curve given locally by f(x,y)=xy has two smooth branches {x=0} and {y=0} meeting at the origin; topologically the link is two arcs crossing and the singularity can be resolved by a single blowup.

 

 

 

 

## Misapplication

Misapplication

Calling a tacnode or a cusp a double point; these have higher-order tangency or a single branch with self-accumulation and are not transverse two-branch crossings.

 

 

 

 

 





## Consequence

Consequence

Local computations (intersection numbers, delta-invariant contributions) and simple topological descriptions apply; deformation theory typically allows smoothing into two distinct nearby points or resolving by a single simple operation.

 

 

 

 

## Reversal

Reversal

If the two local branches share the same tangent direction (are tangent) the singularity is not a double point but a higher tangency singularity such as a tacnode.

 

 

 

 

 





## Boundary

Boundary

Applies to isolated, transverse two-branch singularities on curves or hypersurfaces; excludes non-isolated self-intersections, higher multiplicity singularities, and singularities arising from a single branch (cusps).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes with 'tacnode' and 'cusp' for informal descriptions of 'where two things meet'; the double point emphasizes transverse, multiplicity-two crossing, whereas tacnode and cusp emphasize tangency or single-branch degeneracy.

 

 

 

 

 





## Synthesis

Synthesis

A double point is the simplest isolated crossing singularity: two distinct smooth local branches intersect with distinct tangent directions, locally equivalent to xy=0, admitting elementary resolution and predictable local invariants.