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=blu

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, MR0130218, http://retro.seals.ch/digbib/view?rid=ensmat-001:1959:5::55 
• Viray, Bianca (2009), The Hasse principle for Chatelet surfaces in characteristic 2, arXiv:0902.3644