Conic bundle
In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form:
- [math]\displaystyle{ X^2 + aXY + b Y^2 = P (T).\, }[/math]
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 [math]\displaystyle{ (a, P) }[/math] in the second Galois cohomology of the field [math]\displaystyle{ k }[/math]. 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:
- [math]\displaystyle{ X^2 - aY^2 = P (T). \, }[/math]
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:
- [math]\displaystyle{ X^2 - aY^2 = P (T) Z^2. \, }[/math]
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 [math]\displaystyle{ T\mapsto T'=1/ T }[/math]), the same fiber (excepted the fibers [math]\displaystyle{ T = 0 }[/math] and [math]\displaystyle{ T '= 0 }[/math]), written as the set of solutions [math]\displaystyle{ X'^2 - aY'^2= P^* (T') Z'^2 }[/math] where [math]\displaystyle{ P^* (T ') }[/math] appears naturally as the reciprocal polynomial of [math]\displaystyle{ P }[/math]. Details are below about the map-change [math]\displaystyle{ [x ':y': z '] }[/math].
The fiber c
Going a little further, while simplifying the issue, limit to cases where the field [math]\displaystyle{ k }[/math] is of characteristic zero and denote by [math]\displaystyle{ m }[/math] any integer except zero. Denote by P(T) a polynomial with coefficients in the field [math]\displaystyle{ k }[/math], of degree 2m or 2m − 1, without multiple roots. Consider the scalar a.
One defines the reciprocal polynomial by [math]\displaystyle{ P^*(T')=T^{2m}P(1/ T) }[/math], and the conic bundle Fa,P as follows:
Definition
[math]\displaystyle{ F_{a,P} }[/math] is the surface obtained as "glueing" of the two surfaces [math]\displaystyle{ U }[/math] and [math]\displaystyle{ U' }[/math] of equations
- [math]\displaystyle{ X^2 - aY^ 2 = P (T) Z^2 }[/math]
and
- [math]\displaystyle{ X '^2 - aY'^2 = P (T ') Z'^ 2 }[/math]
along the open sets by isomorphism
- [math]\displaystyle{ x '= x, y' = y, }[/math] and [math]\displaystyle{ z '= z t^m }[/math].
One shows the following result:
Fundamental property
The surface Fa,P is a k smooth and proper surface, the mapping defined by
- [math]\displaystyle{ p: U \to P_{1, k} }[/math]
by
- [math]\displaystyle{ ([x:y:z],t)\mapsto t }[/math]
and the same on [math]\displaystyle{ U ' }[/math] gives to Fa,P a structure of conic bundle over P1,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.
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