  We study ultrafilters on $\omega^2$ produced by forcing with the quotient of
$\scr P(\omega^2)$ by the Fubini square of the Fr\'echet filter on $\omega$. We
show that such an ultrafilter is a weak P-point but not a P-point and that the
only non-principal ultrafilters strictly below it in the Rudin-Keisler order
are a single isomorphism class of selective ultrafilters. We further show that
it enjoys the strongest square-bracket partition relations that are possible
for a non-P-point. We show that it is not basically generated but that it
shares with basically generated ultrafilters the property of not being at the
top of the Tukey ordering. In fact, it is not Tukey-above
$[\omega_1]^{<\omega}$, and it has only continuum many ultrafilters Tukey-below
it. A tool in our proofs is the analysis of similar (but not the same)
properties for ultrafilters obtained as the sum, over a selective ultrafilter,
of non-isomorphic selective ultrafilters.
