  We present numerical simulations of how a 120 M$_\odot$ primordial star
regulates star formation in nearby cosmological halos at $z \sim$ 20 by
photoevaporation. Our models include nine-species primordial chemistry and
self-consistent multifrequency conservative transfer of UV photons with all
relevant radiative processes. Whether or not new stars form in halos clustered
around a Population III star ultimately depends on their core densities and
proximity to the star. Diffuse halos with central densities below 2 - 3
cm$^{-3}$ are completely ionized and evaporated anywhere in the cluster.
Evolved halos with core densities above 2000 cm$^{-3}$ are impervious to both
ionizing and Lyman-Werner flux at most distances from the star and collapse as
quickly as they would in its absence. Star formation in halos of intermediate
density can be either promoted or suppressed depending on how the I-front
remnant shock compresses, deforms and enriches the core with H$_2$. We find
that the 120 M$_\odot$ star photodissociates H$_2$ in most halos in the cluster
but that catalysis by H- restores it a few hundred kyr after the death of the
star, with little effect on star formation. Our models exhibit significant
departures from previous one-dimensional spherically-symmetric simulations,
which are prone to serious errors due to unphysical geometric focusing effects.
