##  [Node](/node-1) 

 Definition

An isolated ordinary double point of a curve or surface where locally two smooth branches meet and cross transversely; algebraically it is a point of multiplicity two with two distinct tangent directions.

 

 

 

 

 

 





## Principle

Principle

A node is characterized by multiplicity two and a nondegenerate quadratic part of the local equation that factors into two distinct linear factors over the residue field, yielding two distinct tangent lines or directions.

 

 

 

 

 





## Demonstration

Demonstration

In the plane the simplest local model is xy=0: two coordinate lines crossing at the origin. A plane curve with local equation xy + higher-order terms has an ordinary node at the origin if the quadratic part is nondegenerate.

 

 

 

 

## Misapplication

Misapplication

Calling any double point a node even when the quadratic part is a square (which would be a cusp) or when branches are nonreduced; alternatively, confusing a node with a transverse intersection of distinct components at a non-isolated singular set.

 

 

 

 

 





## Consequence

Consequence

Nodes are among the mildest singularities: they are stable under small perturbations, normalize to two distinct smooth branches, and contribute predictable corrections to invariants (e.g., δ and arithmetic genus).

 

 

 

 

## Reversal

Reversal

A cusp or worse singularity where branches do not cross transversely but are tangent or more degenerate.

 

 

 

 

 





## Boundary

Boundary

Refers to ordinary double points only; excludes higher multiplicity singularities, tacnodes (higher-order tangency), and nonreduced double structures. Behavior may differ in positive characteristic if factorization changes.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes with the broader phrase 'ordinary double point' and with intersection-theoretic transverse crossings; in some contexts 'node' emphasizes planarity or isolatedness while 'double point' can be more general.

 

 

 

 

 





## Synthesis

Synthesis

A node is an ordinary double point where two smooth local branches meet with distinct tangents — algebraically signaled by a nondegenerate quadratic part that splits into two linear factors and geometrically by a transverse crossing.