Samiksha Jaiswal (Editor)

Conic bundle

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

Contents

X 2 + a X Y + 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 − a Y 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 − a Y 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 ↦ T ′ = 1 T ), the same fiber (excepted the fibers T = 0 and T ′ = 0 ), written as the set of solutions X ′ 2 − a Y ′ 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 2 m 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 − a Y 2 = P ( T ) Z 2

and

X ′ 2 − a 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 → P 1 , k

by

( [ x : y : z ] , t ) ↦ t

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

References

Conic bundle Wikipedia
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