Conic bundle

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

$X^2+aXY+b{Y}^2=P(T).$

Conic bundles can be considered as a Severi–Brauer surface, or, more precisely, a Châtelet surface. This can be a double covering of a ruled surface. Through an isomorphism, it can be associated with the symbol $(a,P)$ in the second Galois cohomology of the field $k$. 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 an alteration, as such:

$X^2-a{Y}^2=P(T).$

In a second step, it should be placed in a projective space in order to complete the surface at infinity.

To achieve this, we write the equation in homogeneous coordinates and express 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 (smooth and proper), and then glue it to infinity by a change of classical maps.

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

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 roots. Consider the scalar a.

One defines the reciprocal polynomial by $P^{*}(T^{\prime })=T^{2m}P(1/T)$, and the conic bundle F_a,P as follows:

Definition

$F_{a,P}$ is the surface obtained as "glueing" of the two surfaces $U$ and $U^{\prime }$ of equations

$X^2-a{Y}^2=P(T)Z^2$

and

$X{\text{'}}^2-aY{\text{'}}^2=P(T^{\prime })Z{\text{'}}^2$

along the open sets by isomorphism

$x^{\prime }=x,y^{\prime }=y,$ and $z^{\prime }=z{t}^m$.

One shows the following result:

Fundamental property

The surface F_a,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 on $U^{\prime }$ 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 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.