Nikita Karpenko and Alexander Merkurjev: Essential dimension of quadrics

Submission: 2002, Jun 13

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension esdim(X) of X, as defined by Oleg Izhboldin, is esdim(X) = dim(X) - i(X) +1, where i(X) is the first Witt index of X (i.e., the Witt index of X over its own function field). Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F(X). Our main theorem states that esdim(X) is less or equal to dim(Y) and that in the case esdim(X) = dim(Y) the quadric X becomes isotropic over the field F(Y). Applying the main theorem to a projective quadric Y, we get a proof of Izhboldin's conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F(X), then esdim(X) is less or equal to esdim(Y), and the equality holds if and only if X is isotropic over F(Y).

2000 Mathematics Subject Classification: 11E04; 14C25

Keywords and Phrases: Quadratic forms, first Witt index, complete varieties, Chow groups, correspondences

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