Conic bundle

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

${\displaystyle X^{2}+aXY+bY^{2}=P(T).\,}$

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 ${\displaystyle (a,P)}$ in the second Galois cohomology of the field ${\displaystyle k}$.

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

${\displaystyle X^{2}-aY^{2}=P(T).\,}$

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

${\displaystyle X^{2}-aY^{2}=P(T)Z^{2}.\,}$

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

The fiber c

Going a little further, while simplifying the issue, limit to cases where the field ${\displaystyle k}$ is of characteristic zero and denote by ${\displaystyle m}$ any integer except zero. Denote by P(T) a polynomial with coefficients in the field ${\displaystyle k}$, of degree 2m or 2m − 1, without multiple root. Consider the scalar a.

One defines the reciprocal polynomial by ${\displaystyle P^{*}(T')=T^{2m}P({\frac {1}{T}})}$, and the conic bundle F_a,P as follows :

Definition

${\displaystyle F_{a,P}}$ is the surface obtained as "gluing" of the two surfaces ${\displaystyle U}$ and ${\displaystyle U'}$ of equations

${\displaystyle X^{2}-aY^{2}=P(T)Z^{2}}$

and

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

along the open sets by isomorphisms

${\displaystyle x'=x,,y'=y,}$ and ${\displaystyle z'=zt^{m}}$.

One shows the following result :

Fundamental property

The surface F_a,P is a k smooth and proper surface, the mapping defined by

${\displaystyle p:U\to P_{1,k}}$

by

${\displaystyle ([x:y:z],t)\mapsto t}$

and the same on ${\displaystyle U'}$ 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 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.