cuspfield

Cuspfield is a term used in some mathematical discussions to denote the local field associated with a cusp singularity on an algebraic curve. In this sense, a cuspfield is the fraction field of the completed local ring at the cusp, capturing the behavior of functions in a small neighborhood of the cusp. The notion is not universally standard, but it is used to describe a concrete local invariant that reflects analytic and algebraic structure near the singular point.

For a classical cusp given by the plane curve y^2 = x^3 over a field k of characteristic zero, the cuspfield is naturally isomorphic to the field of formal Laurent series k((t)) in a local parameter t, via the standard parametrization x = t^2, y = t^3. In this description, the valuation v on k((t)) measures the order of vanishing at t = 0, the valuation ring is k[[t]], and the residue field is k. Thus the cuspfield encapsulates the local analytic geometry of the cusp inside a complete, discretely valued field.

Applications of the cuspfield idea lie in studying deformations and resolutions of singularities, comparing local behavior of different cusps, and analyzing how function fields specialize in families. The concept can extend to higher dimensions where cusp-like singularities occur, although in those contexts the term cuspfield is less standardized and may be described using analogous local-field constructions.

See also cusp singularity, local ring, completion, Laurent series, local field, normalization, and singularity theory.