##  [Multiple Point](/multiple-point-0) 

 Definition

An isolated boundary or image point at which k distinct local branches or sheets of a mapping or of a geometric object meet for some integer k ≥ 3, producing an intersection of multiplicity k rather than a simple transverse double point.

 

 

 

 

 

 





## Principle

Principle

A multiple point is classified by the number of distinct local preimage branches meeting at the image and by their mutual tangency and intersection orders; multiplicity counts geometric sheets rather than algebraic coincidence alone.

 

 

 

 

 





## Demonstration

Demonstration

A plane configuration formed by three distinct smooth arcs or lines crossing at a single point (for example, three lines through the origin with pairwise distinct directions) produces a Triple Multiple Point where three local branches meet at that point.

 

 

 

 

## Misapplication

Misapplication

Using 'Multiple Point' to label any complicated neighborhood or dense self-intersection, or conflating a transverse double point (node) with a genuine multiple point of multiplicity three or more.

 

 

 

 

 





## Consequence

Consequence

Multiple points change local topological type and intersection numbers, affect invariants such as local contribution to genus or linking, and constrain allowable deformations; their resolution often requires blowing up or local normalization.

 

 

 

 

## Reversal

Reversal

The opposite concept is a simple transverse double point (node) or a regular point: situations in which at most two distinct local branches meet transversely or no branching occurs at all.

 

 

 

 

 





## Boundary

Boundary

Pertains to isolated intersections of smooth or analytic branches where the number of distinct local sheets is finite and well-defined; excludes accumulation points of intersections, nonisolated singular loci, and pathological nonmanifold behaviors.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes with notions of 'high-multiplicity singularity' that emphasize algebraic multiplicity of a defining function; here the emphasis is on distinct geometric branches meeting, which may differ from algebraic multiplicity in nonreduced settings.

 

 

 

 

 





## Synthesis

Synthesis

A Multiple Point is an isolated image point where three or more distinct local geometric branches meet; it is measured by branch count and contact orders, has definite topological and enumerative consequences, and is distinct from simple double nodes or nonisolated intersections.