  The generic ultrafilter $\mathcal{G}_2$ forced by
$\mathcal{P}(\omega\times\omega)/($Fin$\otimes$Fin) was recently proved to be
neither maximum nor minimum in the Tukey order of ultrafilters (in a recent
paper of Blass, Dobrinen, and Raghavan), but it was left open where exactly in
the Tukey order it lies. We prove that $\mathcal{G}_2$ is in fact Tukey minimal
over its projected Ramsey ultrafilter. Furthermore, we prove that for each
$k\ge 2$, the collection of all nonprincipal ultrafilters Tukey reducible to
the generic ultrafilter $\mathcal{G}_k$ forced by
$\mathcal{P}(\omega^k)/$Fin$^{\otimes k}$ forms a chain of length $k$.
Essential to the proof is the extraction of a dense subset $\mathcal{E}_k$ from
(Fin$^{\otimes k})^+$ which we prove to be a topological Ramsey space. The
spaces $\mathcal{E}_k$, $k\ge 2$, form a hiearchy of high dimensional Ellentuck
spaces. New Ramsey-classification theorems for equivalence relations on fronts
on $\mathcal{E}_k$ are proved, extending the Pudlak-Rodl Theorem for fronts on
the Ellentuck space, which are applied to find the Tukey structure below
$\mathcal{G}_k$.
