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| 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 curve]]s to corresponding invariants of their [[dual curve]]s. The invariant called the [[genus (mathematics)|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.
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| ==Plücker invariants and basic equations==
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| 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''<sup>*</sup> called the [[dual curve]]. In the correspondence between the projective plane and its dual, points on ''C'' correspond to lines tangent ''C''<sup>*</sup>, so the dual of ''C''<sup>*</sup> can be identified with ''C''.
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| The first two invariants covered by the Plücker formulas are the degree ''d'' of the curve ''C'' and the degree ''d''<sup>*</sup>, classically called the ''class'' of ''C''. Geometrically, ''d'' is the number of times a given line intersects ''C'', including complex points and points at infinity, with multiplicities properly counted. Similarly, ''d''<sup>*</sup> is the number of [[tangent]]s 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 [[singular point of an algebraic variety|singularities]], the first Plücker equation states that
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| :<math>d^* = d(d-1)\,</math>
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| but this must be corrected for singular curves.
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| Of the [[double point]]s of ''C'', let δ be the number that are ordinary, i.e. that have distinct tangents (these are also called ''[[crunode|nodes]]'') or are [[acnode|isolated points]], and let κ be the number that are [[cusp (singularity)|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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| :<math>d^* = d(d-1)-2\delta-3\kappa.\,</math>
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| Similarly, let δ<sup>*</sup> be the number of ordinary double points, and κ<sup>*</sup> the number of cusps of ''C''<sup>*</sup>. Then the second Plücker equation states
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| :<math>\kappa^* = 3d(d-2)-6\delta-8\kappa.\,</math>
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| The geometric interpretation of an ordinary double point of ''C''<sup>*</sup> is a line that is tangent to the curve at two points ([[bitangent|double tangent]]) and the geometric interpretation of a cusp of ''C''<sup>*</sup> is a [[inflection point|point of inflection]] (stationary tangent).
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| The first two Plücker equations have dual versions:
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| :<math>d = d^*(d^*-1)-2\delta^*-3\kappa^*,\,</math>
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| :<math>\kappa = 3d^*(d^*-2)-6\delta^*-8\kappa^*.\,</math>
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| 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''<sup>*</sup>, δ, δ<sup>*</sup>, κ, κ<sup>*</sup>, the remaining three can be computed.
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| Finally, the [[geometric genus|genus]] of ''C'', classically known as the deficiency of ''C'', can be defined as
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| :<math>g={1\over 2}(d-1)(d-2)-\delta-\kappa.</math>
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| This is equal to the dual quantity
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| :<math>g={1\over 2}(d^*-1)(d^*-2)-\delta^*-\kappa^*</math>
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| and is a positive integer.
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| 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.
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| <!-- Work in later, On the other hand assuming ''C'' nonsingular gives the classical genus formula
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| :''g'' = (''d'' − 1)(''d'' − 2)/2. -->
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| ==Non-singular curves==
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| 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:
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| :<math>d^* = d(d-1)\,</math>
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| :<math>\delta^*={1\over 2}d(d-2)(d-3)(d+3)</math>
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| :<math>\kappa^* = 3d(d-2)\,</math>
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| :<math>g={1\over 2}(d-1)(d-2).</math>
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| So, for example, a non-singular [[quartic plane curve]] is of genus 3 and has 28 bitangents and 24 points of inflection.
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| ==Curve types==
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| 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''<sup>*</sup>=2, δ=δ<sup>*</sup>=κ=κ<sup>*</sup>=''g''=0.
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| For curves of degree 3 there are three possible types, given by:<ref>{{cite book |title=Plane Algebraic Curves|first=Harold|last=Hilton|publisher=Oxford|year=1920|page=201
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| |url=http://www.archive.org/details/cu31924001544216}}</ref>
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| {| class="wikitable"
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| |-
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| ! Type !! ''d'' !! ''d''<sup>*</sup> !! δ !! δ<sup>*</sup> !! κ !! κ<sup>*</sup> !! ''g''
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| |-
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| | (i) || 3 || 6 || 0 || 0 || 0 || 9 || 1
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| |-
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| | (ii) || 3 || 4 || 1 || 0 || 0 || 3 || 0
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| |-
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| | (iii) || 3 || 3 || 0 || 0 || 1 || 1 || 0
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| |}
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| 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 curve]]s).
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| For curves of degree 4 there are 10 possible types, given by:<ref>Hilton p. 264</ref>
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| {| class="wikitable"
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| |-
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| ! Type !! ''d'' !! ''d''<sup>*</sup> !! δ !! δ<sup>*</sup> !! κ !! κ<sup>*</sup> !! ''g''
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| |-
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| | (i) || 4 || 12 || 0 || 28 || 0 || 24 || 3
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| |-
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| | (ii) || 4 || 10 || 1 || 16 || 0 || 18 || 2
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| |-
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| | (iii) || 4 || 9 || 0 || 10 || 1 || 16 || 2
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| |-
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| | (iv) || 4 || 8 || 2 || 8 || 0 || 12 || 1
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| |-
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| | (v) || 4 || 7 || 1 || 4 || 1 || 10 || 1
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| |-
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| | (vi) || 4 || 6 || 0 || 1 || 2 || 8 || 1
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| |-
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| | (vii) || 4 || 6 || 3 || 4 || 0 || 6 || 0
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| |-
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| | (viii) || 4 || 5 || 2 || 2 || 1 || 4 || 0
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| |-
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| | (ix) || 4 || 4 || 1 || 1 || 2 || 2 || 0
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| |-
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| | (x) || 4 || 3 || 0 || 1 || 3 || 0 || 0
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| |}
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| ==References==
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| {{reflist}}
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| *{{springer
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| | title=Plücker formulas
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| | id=P/p072900
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| | last=Shokurov
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| | first=V. V.
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| }}
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| *Salmon, George (1879) [http://www.archive.org/details/treatiseonhigher00salmuoft ''A Treatise on the Higher Plane Curves''] pp. 64ff.
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| {{DEFAULTSORT:Plucker Formula}}
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| [[Category:Algebraic curves]]
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Let me initial start by introducing myself. My title is Boyd Butts although it is not the title on my beginning certification. For a while she's been in South Dakota. My working day occupation is a meter reader. To collect cash is what his family and him enjoy.
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