We introduce an ALMA band 3 spectroscopic survey targeting the brightest submillimeter galaxies (SMGs) in the COSMOS field. Here we present the first results based on the 18 primary SMGs that have 870 mu m flux densities of S-870 =12.4-19.3 mJy and are drawn from a parent sample of 260 ALMA-detected SMGs from the AS2COSMOS survey. We detect emission lines in 17 and determine their redshifts to be in the range of z = 2-5 with a median of 3.3 +/- 0.3. We confirm that SMGs with brighter S(870 )are located at higher redshifts. The data additionally cover five fainter companion SMGs, and we obtain line detection in one. Together with previous studies, our results indicate that for SMGs that satisfy our selection, their brightest companion SMGs are physically associated with their corresponding primary SMGs >= 40% of the time, suggesting that mergers play a role in the triggering of star formation. By modeling the foreground gravitational fields, <10% of the primary SMGs can be strongly lensed with a magnification mu > 2. We determine that about 90% of the primary SMGs have lines that are better described by double Gaussian profiles, and the median separation of the two Gaussian peaks is 430 +/- 40 km s(-1). This allows estimates of an average baryon mass, which, together with the line dispersion measurements, puts our primary SMGs on the similar mass-sigma correlation found on local early-type galaxies. Finally, the number density of our z > 4 primary SMGs is found to be 11 x 10(-6 )cMpc(-3), suggesting that they can be the progenitors of z similar to 3-4 massive quiescent galaxies.