Plücker formulas

The Plucker formulas combine certain invariants of algebraic curves and their dual curves. In addition, they can be related to the common topological invariant of the gender of the curve. They were introduced by Julius Plücker in 1834.

Let an algebraic curve C be given by an equation of degree (order) in the complex projective plane . The dual curve is given by the tangents to curve C and is an algebraic curve in the dual projective plane . Let its degree be (also called the class of curve C). The degree d results from the number of points of intersection of a straight line with curve C, whereby the multiplicity of the points of intersection must be taken into account. Complex points and the point at infinity are also taken into account. The degree is equal to the number of straight lines through a point that are tangents to curve C (also counted with multiplicities here). For a conic section is . For non-singular curves C: ${\displaystyle d}$ ${\displaystyle d^{*}}$ ${\displaystyle d^{*}}$ ${\displaystyle d=d^{*}=2}$

${\displaystyle d^{*}=d(d-1)\,}$.

For the singularities, for the sake of simplicity, only colons (two different tangents), whose number is, and points (only one tangent), whose number is, are considered. Accordingly, in the dual space there are double tangents (dual to the colons) and turning tangents (dual to the tips, the turning tangent touches the curve at the turning points with at least order 3). The Plücker equations then apply: ${\displaystyle \delta }$ ${\displaystyle \kappa }$ ${\displaystyle \delta ^{*}}$ ${\displaystyle \kappa ^{*}}$

${\displaystyle d^{*}=d(d-1)-2\delta -3\kappa }$

${\displaystyle \kappa ^{*}=3d(d-2)-6\delta -8\kappa \,}$

and vice versa:

${\displaystyle d=d^{*}(d^{*}-1)-2\delta ^{*}-3\kappa ^{*}\,}$

${\displaystyle \kappa =3d^{*}(d^{*}-2)-6\delta ^{*}-8\kappa ^{*}\,}$.

The four equations are not independent; any three equations result in the fourth.

With the formulas, Plücker was able to predict, for example, that a cube (d = 3) without singularities always has nine turning lines ( ) and thus nine turning points (six of which are in the complex). ${\displaystyle \kappa ^{*}=9}$

Finally, one can define the topological gender of C:

${\displaystyle g={1 \over 2}(d-1)(d-2)-\delta -\kappa }$

or with the dual invariants:

${\displaystyle g={1 \over 2}(d^{*}-1)(d^{*}-2)-\delta ^{*}-\kappa ^{*}}$.

The formula for gender comes from Alfred Clebsch ( Bernhard Riemann had previously introduced the topological gender of associated Riemann surfaces). With the formula for gender, the number of possible singularities can be further restricted.

literature

• Phillip Griffiths , Joe Harris : Principles of algebraic geometry, Wiley 1978, 1994, p. 263ff

Web links

• Plücker formulas, Encyclopedia of Mathematics, Springer

Individual evidence

1. ^ Plücker, System of analytical geometry based on new ways of looking at things, 1835
2. ↑ The peaks are also referred to as return points, for example Felix Klein, Development of Mathematics in the 19th Century, Springer, Volume 1, p. 124
3. ^ Clebsch, Paul Gordan, Theory of Abelian Functions, Leipzig 1866