  We compare the BCS and Overhauser effect as competing mechanisms for the
destabilization of the quark Fermi surface at asymptotically large chemical
potential, for the special case of 2 space and 1 time dimension. We use the
framework of perturbative one-gluon exchange, which dominates the pairing at
$\mu/g^2\gg 1$. With screening in matter, we show that in the weak coupling
limit the Overhauser effect can compete with the BCS effect only for a
sufficiently large number of colors. Both the BCS and the Overhauser gaps are
of order $g^{4}/\mu$ in Landau gauge.
