Conic bundle: Difference between revisions
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== A naive point of view == |
== A naive point of view == |
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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 harmless alteration, resulting in a more straightforward and uncomplicated expression such as: |
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: <math> X^2 - aY^2 = P (T). \, </math> |
: <math> X^2 - aY^2 = P (T). \, </math> |
Revision as of 00:48, 8 August 2023
This article's tone or style may not reflect the encyclopedic tone used on Wikipedia. (June 2009) |
In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form
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 in the second Galois cohomology of the field .
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
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 harmless alteration, resulting in a more straightforward and uncomplicated expression such as:
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
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 ), the same fiber (excepted the fibers and ), written as the set of solutions where appears naturally as the reciprocal polynomial of . Details are below about the map-change .
The fiber c
Going a little further, while simplifying the issue, limit to cases where the field is of characteristic zero and denote by any integer except zero. Denote by P(T) a polynomial with coefficients in the field , of degree 2m or 2m − 1, without multiple root. Consider the scalar a.
One defines the reciprocal polynomial by , and the conic bundle Fa,P as follows :
Definition
is the surface obtained as "gluing" of the two surfaces and of equations
and
along the open sets by isomorphisms
- and .
One shows the following result :
Fundamental property
The surface Fa,P is a k smooth and proper surface, the mapping defined by
by
and the same on gives to Fa,P a structure of conic bundle over P1,k.
See also
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.