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

hCOSMOS: a dense spectroscopic survey of $r\leqslant21.3$ galaxies in the COSMOS field