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

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

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

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

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

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

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

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

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.