Observation of a point source of astrophysical neutrinos would be a smoking gun signature of a cosmic-ray accelerator. While IceCube has recently discovered a diffuse flux of astrophysical neutrinos, no localized point source has been observed. Previous IceCube searches for point sources in the southern sky were restricted by either an energy threshold above a few hundred TeV or poor neutrino angular resolution. Here we present a search for southern sky point sources with greatly improved sensitivities to neutrinos with energies below 100 TeV. By selecting charged-current $ u_{mu}$ interacting inside the detector, we reduce the atmospheric background while retaining efficiency for astrophysical neutrino-induced events reconstructed with sub-degree angular resolution. The new event sample covers three years of detector data and leads to a factor of ten improvement in sensitivity to point sources emitting below 100 TeV in the southern sky. No statistically significant evidence of point sources was found, and upper limits are set on neutrino emission from individual sources. A posteriori analysis of the highest-energy ~100 TeV starting event in the sample found that this event alone represents a $2.8sigma$ deviation from the hypothesis that the data consists only of atmospheric background.
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