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 formula_2 in the second Galois cohomology of the field formula_3.
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.
A naive point of view.
To write correctly a conic bundle, one must first reduce the quadratic form of the left hand side. Thus, after a harmless change, it has a simple expression like
In a second step, it should be placed in a projective space in order to complete the surface "at infinity".
To do this, we write the equation in homogeneous coordinates and expresses the first visible part of the fiber
That is not enough to complete the fiber as non-singular (smooth and proper), and then glue it to infinity by a change of classical maps:
Seen from infinity, (i.e. through the change formula_6), the same fiber (excepted the fibers formula_7 and formula_8), written as the set of solutions formula_9 where formula_10 appears naturally as the reciprocal polynomial of formula_11. Details are below about the map-change formula_12.
The fiber "c".
Going a little further, while simplifying the issue, limit to cases where the field formula_3 is of characteristic zero and denote by formula_14 any integer except zero. Denote by "P"("T") a polynomial with coefficients in the field formula_3, of degree 2"m" or 2"m" − 1, without multiple root. Consider the scalar "a".
One defines the reciprocal polynomial by formula_16, and the conic bundle "F""a","P" as follows :
Definition.
formula_17 is the surface obtained as "gluing" of the two surfaces formula_18 and formula_19 of equations
and
along the open sets by isomorphisms
One shows the following result :
Fundamental property.
The surface "F""a","P" is a "k" smooth and proper surface, the mapping defined by
by
and the same on formula_26 gives to "F""a","P" a structure of conic bundle over "P"1,"k".