A gravitational wave event, S190510g, which was classified as a binary-neutron-star coalescence at the time of preliminary alert, was detected by LIGO/Virgo collaboration on May 10, 2019. At 1.7 hours after the issue of its preliminary alert, we started a target-of-opportunity imaging observation in Y-band to search for its optical counterpart using the Hyper Suprime-Cam (HSC) on the Subaru Telescope. The observation covers a 118.8 deg$^2$ sky area corresponding to 11.6% confidence in the localization skymap released in the preliminary alert and 1.2% in the updated skymap. We divided the observed area into two fields based on the availability of HSC reference images. For the fields with the HSC reference images, we applied an image subtraction technique; for the fields without the HSC reference images, we sought individual HSC images by matching a catalog of observed objects with the PS1 catalog. The search depth is 22.28 mag in the former method and the limit of search depth is 21.3 mag in the latter method. Subsequently, we performed visual inspection and obtained 83 candidates using the former method and 50 candidates using the latter method. Since we have only the 1-day photometric data, we evaluated probability to be located inside the 3D skymap by estimating their distances with photometry of associated extended objects. We found three candidates are likely located inside the 3D skymap and concluded they could be an counterpart of S190510g, while most of 133 candidates were likely to be supernovae because the number density of candidates was consistent with the expected number of supernova detections. By comparing our observational depth with a light curve model of such a kilonova reproducing AT2017gfo, we show that early-deep observations with the Subaru/HSC can capture the rising phase of blue component of kilonova at the estimated distance of S190510g (~230 Mpc).
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