Nonstandard quadratic forms over rational function fields (Q2684793)

Let \(F\) be a field of characteristic different from \(2\), \(L=F(t_1,\dots,t_m)\) the field of rational functions in \(m\) algebraically independent variables over \(F\), \(\varphi\) an anisotropic form over \(L\) and \(\psi_1,\dots,\psi_n\) anisotropic forms over \(F\). We say that \(\varphi\) is stable with respect to \(\{\psi_1,\dots,\psi_n\}\) if \(\varphi_{K(t_1,\dots,t_m)}\) is anisotropic for any extension \(K/F\) for which \((\psi_i)_K\) is anisotropic for \(i\in \{1,\dots,n\}\). The form \(\varphi\) is called nonstandard (for the extension \(L/F\)) if it is stable with respect to some \(\{\psi_1,\dots,\psi_n\}\), and for the every discrete \(F\)-valuation \(\mathfrak{v}\) on \(L\), \(\varphi_{L_\mathfrak{v}}\) is isotropic (where \(L_\mathfrak{v}\) denotes the completion of \(L\) with respect to this valuation). Let \(\mathcal{X}\) be a \(d\)-dimensional variety over an algebraically closed field \(k\). The author conjectures that for any \(m,d\in \mathbb{N}\), there exists a nonstandard form of dimension \(2^{m+d-1}+1\) for the extension \(k(\mathcal{X})(t_1,\dots,t_m)/k(\mathcal{X})\). The author proves this conjecture in three different cases: (1) \(d=2,m=1\), (2) \(d=3, m=1\), and (3) \(d=1, m=2\). Curiously enough, the case of \(d=m=1\) remains open.