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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 . 抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「conic bundle」の詳細全文を読む スポンサード リンク
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