  How, when and where the first stars formed are fundamental questions
regarding the epoch of Cosmic Dawn. A second order effect in the fluid
equations was recently found to make a significant contribution: an offset
velocity between gas and dark matter, the so-called streaming velocity.
Previous simulations of a limited number of low-mass dark matter haloes suggest
that this streaming velocity can delay the formation of the first stars and
decrease halo gas fractions and the halo mass function in the low mass regime.
However, a systematic exploration of its effects in a large sample of haloes
has been lacking until now. In this paper, we present results from a set of
cosmological simulations of regions of the Universe with different streaming
velocities performed with the moving mesh code AREPO. Our simulations have very
high mass resolution, enabling us to accurately resolve minihaloes as small as
$10^5 \: {\rm M_{\odot}}$. We show that in the absence of streaming, the least
massive halo that contains cold gas has a mass $M_{\rm halo, min} = 5 \times
10^{5} \: {\rm M_{\odot}}$, but that cooling only becomes efficient in a
majority of haloes for halo masses greater than $M_{\rm halo,50\%} = 1.6 \times
10^6 \: {\rm M_{\odot}}$. In regions with non-zero streaming velocities,
$M_{\rm halo, min}$ and $M_{\rm halo,50\%}$ both increase significantly, by
around a factor of a few for each one sigma increase in the value of the local
streaming velocity. As a result, in regions with streaming velocities
$v_\mathrm{stream} \ge 3\,\sigma_\mathrm{rms}$, cooling of gas in minihaloes is
completely suppressed, implying that the first stars in these regions form
within atomic cooling haloes.
