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

[math]\displaystyle{ y^2-az^2=P(x), \, }[/math]

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

References

• Châtelet, F. (1959), "Points rationnels sur certaines courbes et surfaces cubiques" (in fr), L'Enseignement mathématique, 2nd ser. 5: 153–170, http://retro.seals.ch/digbib/view?rid=ensmat-001:1959:5::55, retrieved 2009-06-23 
• Viray, Bianca (2012), "Failure of the Hasse principle for Châtelet surfaces in characteristic 2", Journal de Théorie des Nombres de Bordeaux 24 (1): 231–236, doi:10.5802/jtnb.794, https://jtnb.cedram.org/item?id=JTNB_2012__24_1_231_0