View source for Plücker formulas

<!--
p0729001.png
$#A+1 = 42 n = 0
$#C+1 = 42 : ~/encyclopedia/old_files/data/P072/P.0702900 Pl\AGucker formulas
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

Formulas relating external characteristics, i.e. ones corresponding to projective imbeddings, and internal characteristics of algebraic varieties (cf. [[Algebraic variety|Algebraic variety]]). The oldest and best known numerical formulas in algebraic geometry are the Plücker formulas for a planar reduced and irreducible curve  $  Z \subset  \mathbf C P  ^ {2} $, 
which has only ordinary double and cuspidal singular points. Let  $  d $
be the degree of the curve  $  Z $, 
i.e. the number of points on  $  Z $
on a straight line in [[General position|general position]] in  $  \mathbf C P  ^ {2} $, 
and let  $  d  ^ {*} $
be the class of  $  Z $, 
i.e. the number of straight lines tangent to  $  Z $
at non-singular points and passing through a given fixed point in general position in  $  \mathbf C P  ^ {2} $. 
The two basic Plücker formulas are

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

$$ \tag{2 }
d  ^ {*}  =  2d + ( 2g- 2)- k,
$$

where  $  g $
is the genus of the non-singular resolution  $  X $
of  $  Z $(
cf. [[Resolution of singularities|Resolution of singularities]]),  $  \delta $
is the number of ordinary double points and  $  k $
is the number of cuspidal points. Formula (1) reduces to the form  $  d  ^ {*} = d( d- 1) $
if  $  Z $
is a non-singular curve.

Other classical Plücker formulas follow from (1) and (2) by duality. If  $  Z $
is not a straight line, then the curve  $  Z  ^ {*} $
dual to  $  Z $
is defined as the closure of the set of tangents to  $  Z $, 
considered as points in the dual plane  $  \mathbf C P  ^ {2*} $. 
A theorem due to J. Plücker [[#References|[3]]] states that the bi-dual curve  $  Z  ^ {**} $
coincides with  $  Z $. 
If it is assumed that  $  Z  ^ {*} $
has only  $  \delta  ^ {*} $
ordinary double points and  $  k  ^ {*} $
cuspidal singular points, the formulas

$$ \tag{1* }
d  =  d  ^ {*} ( d  ^ {*} - 1)- 2 \delta  ^ {*} - 4k  ^ {*} ,
$$

$$ \tag{2* }
d  =  2d  ^ {*} + 2( 2g- 2)- 2k  ^ {*}
$$

are obtained. The number  $  \delta  ^ {*} $
can be interpreted also as the number of bitangents to  $  Z $, 
i.e. straight lines that touch  $  Z $
at precisely two different non-singular points with contact of order two, while  $  k  ^ {*} $
is the number of points of inflection.

The four formulas (1), (2), (1ast), and (2ast) are not independent: the fourth follows from any other three. However, any three of them are independent. They also imply the following formulas:

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

$$ \tag{3* }
k  =  3d  ^ {*} ( d  ^ {*} - 2)- 6 \delta  ^ {*} - 8k  ^ {*} .
$$

These formulas were derived by Plücker together with (1) and (1ast) in 1834–1839.

In the case of a ground field of finite characteristic, the Plücker formulas and the duality theorem are not always correct. For example, in characteristic 2 all the tangents to a conic pass through one point, which is called the strange point on the conic, so the dual curve is a straight line. In characteristic 3 there is a non-singular cubic with only three points of inflection, or even with one (according to the Plücker formulas there should be nine). With the correct interpretation of  $  d  ^ {*} $, 
(1) and (2) remain correct in all characteristics  $  \neq 2 $; 
in characteristic 2 they must be replaced by

$$ \tag{1** }
d  ^ {*}  =  d( d- 1)- 2 \delta - 4k,
$$

$$ \tag{2** }
d  ^ {*}  =  2d + ( 2g- 2)- 2k.
$$

There is a generalization of the Plücker formulas to curves in  $  P  ^ {n} $
with arbitrary singularities [[#References|[2]]], and also to the case of hyperplanes in  $  P  ^ {n} $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Berzolari, "Algebraische Transformationen und Korrespondenzen" , ''Enzyklopaedie der math. Wissenschaften'' , '''3''' : Heft 12 , Teubner (1906) pp. 1787–2218</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.L. Kleiman, "The enumerative theory of singularities" , ''Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School'' , Sijthoff &amp; Noordhoff (1977) pp. 297–396</TD></TR></table>

====Comments====
See also [[Plane real algebraic curve|Plane real algebraic curve]] for the notions of class, degree, dual, etc. of a curve.