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conic bundle : ウィキペディア英語版
conic bundle

In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form
: X^2 + aXY + b Y^2 = P (T).\,
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 (a, P) in the second Galois cohomology of the field 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.
== 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
: X^2 - aY^2 = P (T). \,
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
: X^2 - aY^2 = P (T) Z^2. \,
That is not enough to complete the fiber as non-singular (clean and smooth), and then glue it to infinity by a change of classical maps:
Seen from infinity, (i.e. through the change T\mapsto T'=\frac 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 (':y': z ' ).

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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