A plane curve singularity where two distinct smooth branches meet with higher-order tangency (osculation), so that they share the same tangent direction and meet with intersection multiplicity greater than two; a standard local model is y^2 = x^4 or two branches with contact of order two.

Principle

Higher-order contact between two smooth branches: tangents coincide and the intersection multiplicity exceeds the transverse value, producing degenerate crossing behavior that influences deformation and resolution patterns.

Demonstration

The affine plane curve defined by y^2 − x^4 = 0 has two smooth branches meeting at the origin with second-order tangency; perturbations can split the tacnode into multiple nodes depending on the chosen deformation.

Misapplication

Treating any non-simple crossing as a tacnode — for example, confusing a cusp (a single branch with a point of vanishing derivative) or an ordinary double point with a tacnode ignores the distinguishing property of two branches with high-order tangency.

Consequence

Tacnodes carry larger local invariants (e.g., higher δ-type contributions) and require more elaborate local resolution procedures than ordinary double points; their presence affects moduli and smoothing behavior more strongly.

Reversal

If the two branches are transverse (distinct tangents), the singularity is an ordinary double point rather than a tacnode; if there is only one branch with self-singularity, the singularity is a cusp, not a tacnode.

Boundary

Applies to isolated plane curve singularities formed by two branches with shared tangent and contact of order ≥2; excludes single-branch singularities (cusps), non-isolated intersections, and higher-multiplicity clusterings of more than two branches.

Semantic Tension

Mistakenly overlaps with 'cusp' and 'node' in casual usage; tacnode is specifically a two-branch phenomenon characterized by osculation, whereas cusp is single-branch and node is transverse two-branch.

Synthesis

A tacnode is the degenerate two-branch plane singularity where branches osculate: tangents coincide and contact order is elevated, producing characteristic deformation and resolution behavior distinct from nodes and cusps.