  We investigate the Hasse principles for isotropy and isometry of quadratic
forms over finitely generated field extensions with respect to various sets of
discrete valuations. Over purely transcendental field extensions of fields that
satisfy property $\mathscr{A}_i(2)$ for some $i$, we find numerous
counterexamples to the Hasse principle for isotropy with respect to a
relatively small set of discrete valuations. For finitely generated field
extensions $K$ of transcendence degree $r$ over an algebraically closed field
of characteristic $\ne 2$, we use the $2^r$-dimensional counterexample to the
Hasse principle for isotropy due to Auel and Suresh to obtain counterexamples
of lower dimensions with respect to the divisorial discrete valuations induced
by a variety with function field $K$.
