Hessian pair

In mathematics, a Hessian pair or Hessian duad, named for Otto Hesse, is a pair of points of the projective line canonically associated with a set of 3 points of the projective line. More generally, one can define the Hessian pair of any triple of elements from a set that can be identified with a projective line, such as a rational curve, a pencil of divisors, a pencil of lines, and so on.

Definition

If {A, B, C} is a set of 3 distinct points of the projective line, then the Hessian pair is a set {P,Q} of two points that can be defined by any of the following properties:

• P and Q are the roots of the Hessian of the binary cubic form with roots A, B, C.
• P and Q are the two points fixed by the unique projective transformation taking A to B, B to C, and C to A.
• P and Q are the two points that when added to A, B, C form an equianharmonic set (a set of 4 points with cross-ratio a cube root of 1).
• P and Q are the images of 0 and ∞ under the projective transformation taking the three cube roots of 1 to A, B, C.

Examples

Hesse points can be used to solve cubic equations as follows. If A, B, C are three roots of a cubic, then the Hesse points can be found as roots of a quadratic equation. If the Hesse points are then transformed to 0 and ∞ by a fractional linear transformation, the cubic equation is transformed to one of the form x³ = D.

See also

• Glossary of classical algebraic geometry

References

• Edge, W. L. (1978), "Bring's curve", Journal of the London Mathematical Society, 18 (3): 539–545, doi:10.1112/jlms/s2-18.3.539, ISSN 0024-6107, MR 0518240
• Inoue, Naoki; Kato, Fumiharu (2005), "On the geometry of Wiman's sextic", Journal of Mathematics of Kyoto University, 45 (4): 743–757, doi:10.1215/kjm/1250281655, ISSN 0023-608X, MR 2226628