These formulae, and the fact that each of the invariants must be a positive integer, place quite strict limitations on their possible values.
In the correspondence between the projective plane and its dual, points on C correspond to lines tangent C*, so the dual of C* can be identified with C. The first two invariants covered by the Plücker formulas are the degree d of the curve C and the degree d*, classically called the class of C. Geometrically, d is the number of times a given line intersects C with multiplicities properly counted.
Similarly, d* is the number of tangents to C that are lines through a given point on the plane; so for example a conic section has degree and class both 2.
Finally, the genus of C, classically known as the deficiency of C, can be defined as This is equal to the dual quantity and is a positive integer.
An important special case is when the curve C is non-singular, or equivalently δ and κ are 0, so the remaining invariants can be computed in terms of d only.
In this case the results are: So, for example, a non-singular quartic plane curve is of genus 3 and has 28 bitangents and 24 points of inflection.