Twocusped

Twocusped is an informal term used in geometry and singularity theory to describe objects that possess exactly two cusp singularities. A cusp is a point where a smooth curve fails to be differentiable in a particular way and locally resembles the standard cusp y^2 = x^3. In this sense, a twocusped curve is smooth at all other points and has precisely two isolated cusps.

In the context of plane algebraic curves, the cusps are isolated singularities, and the local analytic type near each cusp is that of a cusp. The two singular points may lie on the same irreducible component or on separate components, depending on the equation defining the curve. The term twocusped is not a formally fixed category in all sources and may be used variably to denote a two-cusped curve or surface in informal discussions.

Twocusped curves are studied to understand how multiple singularities interact under deformation and to illustrate questions in enumerative geometry and singularity theory. Invariants associated with singularities, such as the delta invariant and Milnor number, contribute to the global topology of the curve, including its genus in the algebraic setting. The two cusps can affect how the curve can be smoothed or perturbed and influence the possible deformation families.

See also: cusp, cuspidal curve, multicusp, singularity theory. References to standard texts on plane curve singularities and deformation theory provide formal treatments of cusps and their interactions.