We present a CO(2-1) and 1240 um continuum survey of 23 debris disks with spectral types B9-G1, observed at an angular resolution of 0.5-1 arcsec with the Atacama Large Millimeter/Submillimeter Array (ALMA). The sample was selected for large infrared excess and age ~10 Myr, to characterize the prevalence of molecular gas emission in young debris disks. We identify three CO-rich debris disks, plus two additional tentative (3-sigma) CO detections. Twenty disks were detected in the continuum at the >3-sigma level. For the 12 disks in the sample that are spatially resolved by our observations, we perform an independent analysis of the interferometric continuum visibilities to constrain the basic dust disk geometry, as well as a simultaneous analysis of the visibilities and broad-band spectral energy distribution to constrain the characteristic grain size and disk mass. The gas-rich debris disks exhibit preferentially larger outer radii in their dust disks, and a higher prevalence of characteristic grain sizes smaller than the blowout size. The gas-rich disks do not exhibit preferentially larger dust masses, contrary to expectations for a scenario in which a higher cometary destruction rate would be expected to result in a larger mass of both CO and dust. The three debris disks in our sample with strong CO detections are all around A stars: the conditions in disks around intermediate-mass stars appear to be the most conducive to the survival or formation of CO.
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