Conic bundle: Difference between revisions
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In [[algebraic geometry]], a '''conic bundle''' is an [[algebraic variety]] that appears as a solution |
In [[algebraic geometry]], a '''conic bundle''' is an [[algebraic variety]] that appears as a solution to a [[Cartesian equation]] of the form: |
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: <math>X^2 + aXY + b Y^2 = P (T).\,</math> |
: <math>X^2 + aXY + b Y^2 = P (T).\,</math> |
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Conic bundles can be considered as either a Severi–Brauer |
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 <math>(a, P)</math> in the second [[Galois cohomology]] of the field <math>k</math> 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 surface]]s 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]].{{Clarify|reason=How is this related to conic bundle?|date=June 2024}} |
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==A point of view== |
==A 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 |
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: |
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: <math> X^2 - aY^2 = P (T). \, </math> |
: <math> X^2 - aY^2 = P (T). \, </math> |
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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: |
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To achieve this, write the equation in [[homogeneous coordinates]] and express the first visible part of the fiber: |
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: <math> X^2 - aY^2 = P (T) Z^2. \, </math> |
: <math> X^2 - aY^2 = P (T) Z^2. \, </math> |
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and |
and |
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:<math>X '^2 - aY'^2 = P (T ') Z'^ 2</math> |
:<math>X '^2 - aY'^2 = P^* (T ') Z'^ 2</math> |
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along the open sets by [[isomorphism]] |
along the open sets by [[isomorphism]] |
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:<math>([x:y:z],t)\mapsto t</math> |
:<math>([x:y:z],t)\mapsto t</math> |
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and the same |
and the same definition applied to <math> U '</math> gives to ''F''<sub>''a'',''P''</sub> a structure of conic bundle over ''P''<sub>1,''k''</sub>. |
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==See also== |
==See also== |
Revision as of 01:19, 30 June 2024
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 to a Cartesian equation of the form:
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 in the second Galois cohomology of the field 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.[clarification needed]
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:
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:
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 roots. 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 isomorphism
- 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 definition applied to 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.