deltainvariant

The delta-invariant, denoted δ, is a classical invariant associated with a singular point of a plane algebraic curve. It measures how far the point is from being nonsingular and, globally, how the curve’s geometric genus changes under normalization.

For a curve C with a singular point P, let OP be the local ring of C at P and let ŌP be its integral closure (the local normalization). The delta-invariant at P is defined by δ(P) = dimk(ŌP / OP), where k is the residue field. If C has several singular points, the total delta-invariant δ(C) is the sum of the δ’s at its singularities. Equivalently, δ(C) equals the difference between the arithmetic genus p_a(C) and the geometric genus p_g(C̃) of the normalization C̃ → C, so δ(C) = p_a(C) − p_g(C̃).

In the special case of plane curve singularities, δ can also be related to other invariants. If the singularity has r branches (components of the germ), the Milnor number μ satisfies μ = 2δ − r + 1. For irreducible plane curve germs, δ also coincides with the number of gaps in the value semigroup of the branch.

Common examples include the node and the cusp, both of which have δ = 1. More complicated singularities have larger δ-values, reflecting greater loss of regularity.

Properties and uses: δ is a nonnegative integer that vanishes exactly at nonsingular points, is additive over singularities, and is upper semicontinuous in families. It plays a central role in the study of deformations, equisingularity, and the genus formula for singular curves, as well as in the classification of curve singularities.