  Quantum phase transition at the saturation field is studied for a class of
frustrated quantum antiferromagnets. The considered models include (i) the
$J_1$-$J_2$ frustrated square-lattice antiferromagnet with $J_2={1/2}J_1$ and
(ii) the nearest-neighbor Heisenberg antiferromagnet on a face centered cubic
lattice. In the fully saturated phase the magnon spectra for the two models
have lines of degenerate minima. Transition into partially magnetized state is
treated via a mapping to a dilute gas of hard core bosons and by complementary
spin-wave calculations. Momentum dependence of the exact four-point boson
vertex removes the degeneracy of the single-particle excitation spectra and
selects the ordering wave-vectors at $(\pi,\pi)$ and $(\pi,0,0)$ for the two
models. The asymptotic behavior of the magnetization curve differs
significantly from that of conventional antiferromagnet in $d$-spatial
dimensions. We predict a unique form for the magnetization curve $\Delta
M=S-M\simeq \mu^{(d-1)/2}(\log\mu)^{(d-1)}$, where $\mu$ is a distance from the
quantum critical point.
