Plücker formula

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In mathematics, a Plücker formula is one of an extensive family of counting formulae, of a type first developed in the 1830s by Julius Plücker, that relate the extrinsic geometry of algebraic curves in projective space to intrinsic invariants such as the genus. They can be applied in either direction, to calculate the genus for example from some geometrical numbers. In the modern approach it is more natural, however, to regard the curve C as given, a linear system of divisors on it as provided, and the extrinsic geometry such as osculation as in some sense controlled by the intrinsic geometry.

Plücker's original work involved the dual curve C* to C. This is defined as the set of tangent lines to the plane curve C, in the complex projective plane. He allowed C to have singular points, in which case C* is better defined as the Zariski closure of the tangent lines at non-singular points of C, in the dual projective plane. Here C* is again a curve, unless C was a line in the first place. Write d for the degree of C, and d* for the degree of C*, classically called the class of C. Geometrically it is the number of tangents to C that are lines through a typical point of the plane not on C; so for example a conic section has degree and class both 2.

If C has no singularities, the first Plücker formula states that

d* = d(d − 1)

and this must be corrected for singular curves. The simplest singularities being double points with multiplicity 2, and cusps with multiplicity 3, the corrected form is

d* = d(d − 1) − 2×(number of double points) − 3×(number of cusps).

One needs double points, at least, to cover all curves. Not all curves 'fit' into the plane without singularities, as the next formula shows.

d* = 2d − χ − (number of cusps).

Here χ is the Euler characteristic 2 − 2g where g is the genus of C. 'Genus' here means geometric genus, i.e. the birational invariant, supposing C is singular. The two formulae together therefore enable one to calculate g given the degree d and the singularities. On the other hand assuming C nonsingular gives the classical genus formula

g = (d − 1)(d − 2)/2.

The RHS runs through a quadratic progression, while the LHS takes on all possible values 0, 1, 2, 3, ... . Therefore the non-singular plane curve case is rather special.