Plücker formula

In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their dual curves. The invariant called the genus, common to both the curve and its dual, is connected to the other invariants by similar formulae. These formulae, and the fact that each of the invariants must be a positive integer, place quite strict limitations on their possible values.

A curve in this context is defined by a non-degenerate algebraic equation in the complex projective plane. Lines in this plane correspond to points in the dual projective plane and the lines tangent to a given algebraic curve C correspond to points in an algebraic curve C^* called the dual curve. 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. (This includes complex points and points at infinity since the curves are taken be subsets of the complex projective plane.) 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. If C has no singularities, the first Plücker equation states that

but this must be corrected for singular curves.

Of the double points of C, let δ be the number that are ordinary, i.e. that have distinct tangents (these are also called nodes) or are isolated points, and let κ be the number that are cusps, i.e. having a single tangent (spinodes). If C has higher order singularities then these are counted as multiple double points according to an analysis of the nature of the singularity. For example an ordinary triple point is counted as 3 double points. Again, complex points and points at infinity are included in these counts. The corrected form is of the first Plücker equation is

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