Difference between revisions of "Plücker formulas"

Formulas relating external characteristics, i.e. ones corresponding to projective imbeddings, and internal characteristics of algebraic varieties (cf. 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 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), $\delta$ 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 [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 [2], and also to the case of hyperplanes in $P ^ {n}$.

References

 [1] L. Berzolari, "Algebraische Transformationen und Korrespondenzen" , Enzyklopaedie der math. Wissenschaften , 3 : Heft 12 , Teubner (1906) pp. 1787–2218 [2] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001 [3] S.L. Kleiman, "The enumerative theory of singularities" , Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School , Sijthoff & Noordhoff (1977) pp. 297–396