A bound for the index of a quadratic form after scalar extension to the function field of a quadric
Abstract
Let q be an anisotropic quadratic form defined over a general field F. In this article, we formulate a new upper bound for the isotropy index of q after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of a celebrated bound established in earlier work of Karpenko-Merkurjev and Totaro; on the other, it is a direct generalization of Karpenko's theorem on the possible values of the first higher isotropy index. We prove its validity in two important cases: (i) the case where char(F) ≠ 2, and (ii) the case where char(F) = 2 and q is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic-geometric, and the second being purely algebraic.