A bound for the index of a quadratic form after scalar extension to the function field of a quadric

TL;DR

This paper introduces a new upper bound for the isotropy index of an anisotropic quadratic form after scalar extension to the function field of a quadric, refining previous bounds and generalizing earlier theorems.

Contribution

It formulates a novel upper bound applicable in characteristic not 2 and for quasilinear forms in characteristic 2, using distinct algebraic-geometric and algebraic methods.

Findings

Refines the Karpenko-Merkurjev and Totaro bound.

Generalizes Karpenko's theorem on higher isotropy indices.

Valid in both characteristic not 2 and characteristic 2 for quasilinear forms.

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 $\mathrm{char}(F) \neq 2$, and (ii) the case where $\mathrm{char}(F) = 2$ 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.