“Conic bundle”的意思、由来-开放百科全书

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Conic bundle

释义

A naive point of view
The fiber c
Definition Fundamental property
See also
References

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

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

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 (clean and smooth), 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 F_a,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 F_a,P is a k clean and smooth surface, the mapping defined by

and the same on gives to F_a,P a structure of conic bundle over P_1,k.

References

{{cite book

| author = Robin Hartshorne
| year = 1977
| title = Algebraic Geometry
| publisher = Springer-Verlag
| isbn = 0-387-90244-9
}}

{{cite book

| author = David Cox
| author2=John Little
| author3=Don O'Shea
| year = 1997
| title = Ideals, Varieties, and Algorithms
| edition = second
| publisher = Springer-Verlag
| isbn = 0-387-94680-2
}}

{{cite book

| author = David Eisenbud
| year = 1999
| title = Commutative Algebra with a View Toward Algebraic Geometry
| publisher = Springer-Verlag
| isbn = 0-387-94269-6
}}

2 : Algebraic geometry|Algebraic varieties

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A naive point of view

The fiber c

Definition

Fundamental property

See also

References