Submission: 2006, Jul 27
A projective quadric over a field k of characteristic 2 is called quasilinear if it is nowhere smooth over k. We show that the quadratic Zariski problem has a positive answer for quasilinear quadrics: if X is isotropic over k(Y), Y is isotropic over k(X), and X and Y have the same dimension, then X and Y are birational. Moreover, a quasilinear quadric is ruled if and only if its first Witt index is greater than 1. 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.
2000 Mathematics Subject Classification: Primary 11E04, Secondary 14E05.
Keywords and Phrases: Quadratic forms, ruled varieties, quadratic Zariski problem, quasilinear forms, essential dimension
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