Conic bundle

In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution to a Cartesian equation of the form:

${\displaystyle X^{2}+aXY+bY^{2}=P(T).\,}$

Conic bundles can be considered as either a Severi–Brauer or Châtelet surface. This can be a double covering of a ruled surface. It can be associated with the symbol ${\displaystyle (a,P)}$ in the second Galois cohomology of the field ${\displaystyle k}$ through an isomorphism. In practice, it is more commonly observed as a surface with a well-understood divisor class group, and the simplest cases share with Del Pezzo surfaces the property of being a rational surface. But many problems of contemporary mathematics remain open, notably, for those examples which are not rational, the question of unirationality.