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 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 F_a,P as follows :

Definition

F_a,P is the surface obtained as "gluing" of the two surfaces U and U' of equations

X² − aY² = P(T)Z²

and

X'² − Y'² = P(T')Z'²

along the open sets by isomorphisms

x' = x,,y' = y, and z' = zt^m.

One shows the following result :

Fundamental property

The surface F_a,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 F_a,P a structure of conic bundle over P_1,k.

See also

• Algebraic surface
• Intersection number (algebraic geometry)
• List of complex and algebraic surfaces

References

• Robin Hartshorne (1977). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9. 
• David Cox; John Little, Don O'Shea (1997). Ideals, Varieties, and Algorithms (second edition ed.). Springer-Verlag. ISBN 0-387-94680-2. 
• David Eisenbud (1999). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag. ISBN 0-387-94269-6.