A tacnode is a planar singularity where two smooth branches have a common tangent and osculate to higher order at their contact point; geometrically the branches meet with higher-order tangency rather than transversely.

Principle

Tacnodes arise when the quadratic (and possibly higher) part of the local equation does not split into distinct linear factors but rather produces two distinct smooth branches whose tangent directions coincide; algebraically this corresponds to multiple factors with higher contact order in the local expansion.

Demonstration

A model is given by y^2 = x^4 near the origin: the curve has two branches y=±x^2 which share the tangent y=0 and meet with second-order contact, producing a tacnode rather than a node or cusp.

Misapplication

Confusing a tacnode with a node (transverse crossing) or with a cusp (unibranch tangency) — a tacnode is multibranch with coincident tangents, not a single-branch pointed singularity nor a transverse intersection.

Consequence

Tacnodes are more complicated to resolve than nodes: they typically require multiple blow-ups to separate branches, affect local intersection multiplicities strongly, and change normalization behavior and enumerative counts.

Reversal

A transverse node, where branches meet with distinct tangent directions, or a unibranch cusp, where only one branch is present.

Boundary

Term applies to planar or curve singularities of higher contact order; excludes ordinary nodes, unibranch cusps, and higher-multiplicity nonreduced contacts. In positive characteristic the local factorization and contact orders can behave differently.

Semantic Tension

Tension with terms like 'osculation' and 'tangency order' — tacnode emphasizes coincidence of tangents for distinct branches, while other terms may emphasize multiplicity or unibranchness.

Synthesis

A tacnode is a multibranch planar singularity in which distinct smooth branches share the same tangent and meet with higher-order contact (e.g., y^2 = x^4), necessitating finer resolution than ordinary nodes.