Conic bundle

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

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

Conic bundles can be considered as either a Severi–Brauer or Châtelet surface. This can be a double covering of a ruled surface. It can be associated with the symbol ${\displaystyle (a,P)}$ in the second Galois cohomology of the field ${\displaystyle k}$ through an isomorphism. 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 a transformation, such as:

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

Next steps involve placement in a projective space to complete the surface at infinity, which may be achieved by writing the equation in homogeneous coordinates and expressing 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 ${\displaystyle T\mapsto T'=1/T}$), 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 roots. Consider the scalar a.

One defines the reciprocal polynomial by ${\displaystyle P^{*}(T')=T^{2m}P(1/T)}$, 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 isomorphism

${\displaystyle x'=x,y'=y,}$ 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 definition applied to ${\displaystyle U'}$ 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.