A descent problem for quadratic forms (Q1908590)

For an anisotropic quadratic form \(q\) over a field \(\mathbb{F}\) of characteristic \(\neq 2\), let \(K=F(q)\) be the function field of the projective hypersurface \(X\) of equation \(q (x) = 0\). Suppose a quadratic form \(\varphi\) over \(K\) has all second residues at codimension 1 points of \(X\) equal to zero. The author considers the conjecture that, if \(\dim\varphi < (1/2) \dim q\), then \(\varphi\) is defined over the field \(F\). This has been proved here when \(\dim \varphi \leq 5\), and partially in dimension 6 (for \(\varphi\) an Albert form) and 8 (for \(\varphi\) a form similar to a Pfister form). As an immediate corollary the author obtains all known bounds on the dimensions of quadratic forms of height 2 established earlier by M. Knebusch, R. W. Fitzgerald, J. Hurrelbrink and U. Rehmann, and the author.