  Fluctuations of the 21 cm brightness temperature before the formation of the
first stars hold the promise of becoming a high-precision cosmological probe in
the future. The growth of over densities is very well described by perturbation
theory at that epoch and the signal can in principle be predicted to arbitrary
accuracy for given cosmological parameters. Recently, Tseliakhovich and Hirata
pointed out a previously neglected and important physical effect, due to the
fact that baryons and cold dark matter (CDM) have supersonic relative
velocities after recombination. This relative velocity suppresses the growth of
matter fluctuations on scales $k \sim 10-10^3$ Mpc$^{-1}$. In addition, the
amplitude of the small-scale power spectrum is modulated on the large scales
over which the relative velocity varies, corresponding to $k \sim 0.005-1$
Mpc$^{-1}$. In this paper, the effect of the relative velocity on 21 cm
brightness temperature fluctuations from redshifts $z \geq 30$ is computed. We
show that the 21 cm power spectrum is affected on ${\it most}$ scales. On small
scales, the signal is typically suppressed several tens of percent, except for
extremely small scales ($k \gtrsim 2000$ Mpc$^{-1}$) for which the fluctuations
are boosted by resonant excitation of acoustic waves. On large scales, 21 cm
fluctuations are enhanced due to the non-linear dependence of the brightness
temperature on the underlying gas density and temperature. The enhancement of
the 21 cm power spectrum is of a few percent at $k \sim 0.1$ Mpc$^{-1}$ and up
to tens of percent at $k \lesssim 0.005$ Mpc$^{-1}$, for standard $\Lambda$CDM
cosmology. In principle this effect allows to probe the small-scale matter
power spectrum not only through a measurement of small angular scales but also
through its effect on large angular scales.
