Conic bundle

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

Theoretically, it can be considered as a Severi–Brauer surface, or more precisely as a Châtelet surface. This can be a double covering of a ruled surface. Through an isomorphism, it can be associated with a symbol ${\displaystyle (a,P)}$ in the second Galois cohomology of the field ${\displaystyle k}$.

In fact, it is a surface with a well-understood divisor class group and 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.

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