  A conic bundle or quadric bundle in characteristic 2 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. First, if there
is a rational map from one quasilinear quadric to another of the same
dimension, and also a map back, then the two quadrics are birational. Next, we
determine exactly which quasilinear quadrics are ruled, that is, birational
over the base field to the product of some variety with the projective line.
Both statements are conjectured for quadrics in any characteristic. The proofs
begin by extending Karpenko and Merkurjev's theorem on the essential dimension
of quadrics to arbitrary quadrics (smooth or not) in characteristic 2.
