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We present Atacama Large Millimeter Array CO(3$-$2) and HCO$^+$(4$-$3)observations covering the central $1\rlap{.}'5$$\times$$1\rlap{.}'5$ region ofthe Orion Nebula Cluster (ONC). The unprecedented level of sensitivity($\sim$0.1 mJy beam$^{-1}$) and angular resolution ($\sim$$0\rlap{.}''09\approx 35$ AU) of these line observations enable us to search for gas-diskdetections towards the known positions of submillimeter-detected dust disks inthis region. We detect 23 disks in gas: 17 in CO(3$-$2), 17 in HCO$^+$(4$-$3),and 11 in both lines. Depending on where the sources are located in the ONC, wesee the line detections in emission, in absorption against the warm background,or in both emission and absorption. We spectrally resolve the gas with $0.5$ kms$^{-1}$ channels, and find that the kinematics of most sources are consistentwith Keplerian rotation. We measure the distribution of gas-disk sizes and findtypical radii of $\sim$50-200 AU. As such, gas disks in the ONC are compact incomparison with the gas disks seen in low-density star-forming regions. Gassizes are universally larger than the dust sizes. However, the gas and dustsizes are not strongly correlated. We find a positive correlation between gassize and distance from the massive star $\theta^1$ Ori C, indicating that disksin the ONC are influenced by photoionization. Finally, we use the observedkinematics of the detected gas lines to model Keplerian rotation and infer themasses of the central pre-main-sequence stars. Our dynamically-derived stellarmasses are not consistent with the spectroscopically-derived masses, and wediscuss possible reasons for this discrepancy.

Protoplanetary Disks in the Orion Nebula Cluster: Gas Disk Morphologies and Kinematics as seen with ALMA