Conic bundle

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


 * $$X^2 + aXY + b Y^2 = P (T).\,$$

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 $$(a, P)$$ in the second Galois cohomology of the field $$k$$ 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.

A point of view
In order to properly express a conic bundle, the initial step involves simplifying the quadratic form on the left side. This can be achieved through a transformation, such as:


 * $$ X^2 - aY^2 = P (T). \, $$

Next steps involve placement in a projective space to complete the surface at infinity, which may be achieved by writing the equation in homogeneous coordinates and expressing the first visible part of the fiber:


 * $$ X^2 - aY^2 = P (T) Z^2. \, $$

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 $$ T\mapsto T'=1/ T$$), the same fiber (excepted the fibers $$T = 0$$ and $$T '= 0$$), written as the set of solutions $$X'^2 - aY'^2= P^* (T') Z'^2 $$ where $$P^* (T ')$$ appears naturally as the reciprocal polynomial of $$P$$. Details are below about the map-change $$[x ':y': z ']$$.

The fiber c
Going a little further, while simplifying the issue, limit to cases where the field $$k$$ is of characteristic zero and denote by $$m$$ any integer except zero. Denote by P(T) a polynomial with coefficients in the field $$k$$, of degree 2m or 2m &minus; 1, without multiple roots. Consider the scalar a.

One defines the reciprocal polynomial by $$P^*(T')=T^{2m}P(1/ T)$$, and the conic bundle Fa,P as follows:

Definition
$$F_{a,P}$$ is the surface obtained as "gluing" of the two surfaces $$U$$ and  $$U'$$ of equations


 * $$ X^2 - aY^ 2 = P (T) Z^2$$

and


 * $$X '^2 - aY'^2 = P^* (T ') Z'^ 2$$

along the open sets by isomorphism


 * $$x '= x, y' = y, $$ and $$z '= z t^m$$.

One shows the following result:

Fundamental property
The surface Fa,P is a k smooth and proper surface, the mapping defined by


 * $$p: U \to P_{1, k}$$

by


 * $$([x:y:z],t)\mapsto t$$

and the same definition applied to $$ U '$$ gives to Fa,P a structure of conic bundle over P1,k.