Kernighan–Lin algorithm

In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form

$X^2+aXY+b{Y}^2=P(T).$

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

$X^2-a{Y}^2=P(T).$

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

$X^2-a{Y}^2=P(T)Z^2.$

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 $T\mapsto T^{\prime }=\frac{1}{T}$), the same fiber (excepted the fibers $T=0$ and $T^{\prime }=0$), written as the set of solutions $X{\text{'}}^2-aY{\text{'}}^2=P^{*}(T^{\prime })Z{\text{'}}^2$ where $P^{*}(T^{\prime })$ appears naturally as the reciprocal polynomial of $P$. Details are below about the map-change $[x^{\prime }:y^{\prime }:z^{\prime }]$.

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 $P^{*}(T^{\prime })=T^{2m}P(\frac{1}{T})$, 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 $U^{\prime }$ of equations

$X^2-a{Y}^2=P(T)Z^2$

and

$X{\text{'}}^2-Y{\text{'}}^2=P(T^{\prime })Z{\text{'}}^2$

along the open sets by isomorphisms

$x^{\prime }=x,,y^{\prime }=y,$ and $z^{\prime }=z{t}^m$.

One shows the following result :

Fundamental property

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

$p:U\to P_{1,k}$

by

$([x:y:z],t)\mapsto t$

and the same on $U^{\prime }$ gives to F_a,P a structure of conic bundle over P_1,k.

See also

• Algebraic surface
• Intersection number (algebraic geometry)
• List of complex and algebraic surfaces

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534