In mathematics, the syzygetic pencil or Hesse pencil, named for

Otto Hesse Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked mainly on algebraic invariants, and geometry. The Hessian matrix, the Hesse nor ...

, is a

pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail ...

(one-dimensional family) of cubic plane

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

s in the

complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...

, defined by the equation :

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

Each curve in the family is determined by a pair of parameter values (

\lambda,\mu

) (not both zero) and consists of the points in the plane whose

homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...

(x,y,z)

satisfy the equation for those parameters. Multiplying both

\lambda

and

\mu

by the same

scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...

does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either

\lambda

\mu

(but not both) to be set to zero. Each curve in the pencil passes through the nine points of the

whose

are some permutation of 0, –1, and a

cube root of unity In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the onl ...

. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are equivalent in pairs giving only nine points. The family of cubics through these nine points forms the Hesse pencil. More generally, one can replace the complex numbers by any field containing a cube root of unity and define the Hesse pencil over this field to be the family of cubics through these nine points. The nine common points of the Hesse pencil are the

inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...

s of each of the cubics in the pencil. Any line that passes through at least two of these nine points passes through exactly three of them; the nine points and twelve lines through triples of points form the

Hesse configuration In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by , is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be realized in the complex projective plane as ...

. Every elliptic curve is

birationally equivalent In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...

to a curve of the Hesse pencil; this is the

Hessian form of an elliptic curve In geometry, the Hessian curve is a plane curve similar to folium of Descartes. It is named after the German mathematician Otto Hesse. This curve was suggested for application in elliptic curve cryptography, because arithmetic in this curve represe ...

. However, the parameters (

\lambda,\mu

) of the Hessian form may belong to an

extension field In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...

of the field of definition of the original curve.

References

* *{{Citation , last1=Grove , first1=Charles Clayton , authorlink = Charles Clayton Grove , title=The syzygetic pencil of cubics with a new geometrical development of its Hesse Group , url=https://archive.org/details/syzygeticpencilo00grovrich , publisher=Baltimore, Md. , year=1906 Elliptic curves