Journal of Algebraic Geometry

Author(s): Burt Totaro
Journal: J. Algebraic Geom. 17 (2008), 577-597.
Posted: March 13, 2008
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Abstract: A conic bundle or quadric bundle in characteristic

can have generic fiber which is nowhere smooth over the function field of the base variety; in that case, the generic fiber is called a quasilinear quadric. We solve some of the main problems of birational geometry for quasilinear quadrics, which remain open for quadrics in characteristic not

: when are two quadrics birational, and when is a quadric ruled over the base field? The proofs begin by extending Karpenko and Merkurjev's theorem on the essential dimension of quadrics to arbitrary quadrics (smooth or not) in characteristic

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