cusps of C*. Then the second Plücker equation states The geometric interpretation of an ordinary double point of C* is a line that is tangent to the curve at two points (double tangent) and the geometric interpretation of a cusp of C* is a point of inflection (stationary tangent). Consider for instance, the case of a smooth cubic: The above formula shows that it has inflections. If the cubic degenerates and gets a double point, then 6 points converge to the singular point and only 3 inflection remain along the singular curve. If the cubic degenerates and gets a cusp then only one inflection remains. Note that the first two Plücker equations have dual versions: The four equations given so far are, in fact, dependent, so any three may be used to derive the remaining one. From them, given any three of the six invariants, d, d*, δ, δ*, κ, κ*, the remaining three can be computed. 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. Altogether there are four independent equations in 7 unknowns, and with them any three of these invariants can be used to compute the remaining four. Non-singular curves 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. Curve types Curves are classified into types according to their Plücker invariants. The Plücker equations together with the restriction that the Plücker invariants must all be natural numbers greatly limits the number of possible types for curves of a given degree. Curves which are projectively equivalent have the same type, though curves of the same type are not, in general, projectively equivalent. Curves of degree 2, conic sections, have a single type given by d=d*=2, δ=δ*=κ=κ*=g=0. For curves of degree 3 there are three possible types, given by: Curves of types (ii) and (iii) are the rational cubics and are call nodal and cuspidal respectively. Curves of type (i) are the nonsingular cubics (elliptic curves). For curves of degree 4 there are 10 possible types, given by: References Salmon, George (1879)