Châtelet surface

Some snapshots showing the real points of the Châtelet surface with P(x)=x^3-5*x^2-6*x. Axis: x=red, y=yellow, z=blue

In algebraic geometry, a Châtelet surface is a rational surface studied by Châtelet (1959) given by an equation

y^2-az^2=P(x), \,

where P has degree 3 or 4. They are conic bundles.

References

Châtelet, F. (1959), "Points rationnels sur certaines courbes et surfaces cubiques", L'Enseignement Mathématique. Revue Internationale. IIe Série 5: 153–170, MR 0130218
Viray, Bianca (2009), The Hasse principle for Chatelet surfaces in characteristic 2, arXiv:0902.3644