Bundle of hesse

In mathematics, a syzygytic beam or a Hesse beam (after the German mathematician Ludwig Otto Hesse ) is a bundle (one-dimensional family) of flat cubic elliptic curves on a complex projective plane that satisfy the equation

\lambda (x^{3}+y^{3}+z^{3})+\mu xyz=0.

{\ displaystyle \ lambda (x ^ {3} + y ^ {3} + z ^ {3}) + \ mu xyz = 0.}

{\ displaystyle \ lambda (x ^ {3} + y ^ {3} + z ^ {3}) + \ mu xyz = 0.}

Each curve in the family is determined by a pair of parameters ( $\lambda ,\mu$ ${\ displaystyle \ lambda, \ mu}$ ${\ displaystyle \ lambda, \ mu}$ ) (not equal to zero at the same time) and consists of points on the plane, homogeneous coordinates $(x,y,z)$ ${\ displaystyle (x, y, z)}$ $(x, y, z)$ which satisfy the equation for these parameters. Multiplication of both parameters $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ and $\mu$ ${\ displaystyle \ mu}$ $\ mu$ by the same scalar value does not change the curve, so that there is only one degree of freedom, but the two-parameter form given above allows either $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ either $\mu$ ${\ displaystyle \ mu}$ $\ mu$ (but not both) take a zero value.

Each beam curve passes through nine points of the complex projective plane , the homogeneous coordinates of which are some permutations of 0, −1 and the cubic root of one . There are three roots of one and six permutations for each root, which gives 18 variants of homogeneous coordinates, but they are pairwise equivalent, so we get nine points. The family of cubic curves through these nine points forms a bundle of Hesse. More generally, you can replace complex numbers with any field containing cubic roots of one, and define a Hessian sheaf over this field as a family of cubic lines passing through these nine points.

The nine common points of the Hessian beam are the inflection points of each curve of the beam. Any line passing at least through a pair of these nine points contains exactly three of them. Nine points and twelve straight lines through triples of points form the Hesse configuration .

Any elliptic curve is birationally equivalent to a curve from the Hesse beam. This is the . However, the parameters ( $\lambda ,\mu$ ${\ displaystyle \ lambda, \ mu}$ ${\ displaystyle \ lambda, \ mu}$ ) of the Hessian form may belong to the extension of the field on which the initial curve is given.

Notes

Literature

Michela Artebani, Igor Dolgachev. The Hesse pencil of plane cubic curves // L'Enseignement Mathématique. Revue Internationale. 2e Série. - 2009. - Vol. 55 , no. 3 - p . 235-273 . - ISSN 0013-8584 . - DOI : 10.4171 / lem / 55-3-3 . - arXiv : math / 0611590 .
Charles Clayton Grove. Hesse Group. - Baltimore, Md., 1906.

Bundle of hesse

Notes

Literature

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