hCOSMOS: a dense spectroscopic survey of $r\leqslant21.3$ galaxies in
  the COSMOS field
  We describe the hCOSMOS redshift survey of the COSMOS field conducted with
the Hectospec spectrograph on the MMT. In the central 1~deg$^2$, the hCOS20.6
subset of the survey is $>90\%$ complete to a limiting $r=20.6$. The hCOSMOS
survey includes 1701 new redshifts in the COSMOS field. We also use the total
of 4362 new and remeasured objects to derive the age sensitive D$_n4000$ index
over the entire redshift interval $0.001\lesssim z\lesssim0.6$. For $85\%$ of
the quiescent galaxies in hCOS20.6, we measure the central line-of-sight
velocity dispersion. To explore potential uses of this survey, we combine
previously measured galaxy sizes, profiles and stellar masses with the
spectroscopy. The comparison reveals the known relations among structural,
kinematic, and stellar population properties. We also compare redshift and
D$_n4000$ distributions of hCOS20.6 galaxies with SHELS; a complete
spectroscopic survey of 4~deg$^2$ observed to the same depth. The redshift
distributions in the two fields are very different but the D$_n4000$
distribution is remarkably similar. The relation between velocity dispersion
and stellar mass for massive hCOS20.6 galaxies is consistent with the local
relation from SDSS. Using measured velocity dispersions, we test a photometric
proxy calibrated to galaxies in the local universe. The systematic differences
between the measured and photometric proxy velocity dispersions are correlated
with galaxy dynamical and stellar population properties highlighting the
importance of direct spectroscopic measurements.
