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+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 $(a,P)$ 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}-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 (smooth and proper), and then glue it to infinity by a change of classical maps:

Seen from infinity, (i.e. through the change $T\mapsto T'={\frac {1}{T}}$ ), 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

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

and

X'^{2}-aY'^{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 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 $U'$ gives to F_a,P a structure of conic bundle over P_1,k.

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.

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