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 [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  1, without multiple root. Consider the scalar a.

One defines the reciprocal polynomial by P^*(T')=T^{2m}P(\frac 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 - Y'^2 = P (T ') Z'^ 2

along the open sets by isomorphisms

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

One shows the following result :

Fundamental property

The surface Fa,P is a k clean and smooth surface, the mapping defined by

p:  U \to P_{1, k}

by

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

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

See also

References