  We discuss the geometry and arithmetic of higher-dimensional analogues of
Ch\^atelet surfaces; namely, we describe the structure of their Brauer and
Picard groups and show that they can violate the Hasse principle. In addition,
we use these varieties to give straightforward generalizations of two recent
results of Poonen. Specifically, we prove that, assuming Schinzel's hypothesis,
the non-m^{th} powers of a number field are diophantine. Also, given a global
field k such that Char(k) = p or k contains the p^{th} roots of unity, we
construct a (p+1)-fold that has no k-points and no \'etale-Brauer obstruction
to the Hasse principle.
