We present SN 2020jfo, a Type IIP supernova in the nearby galaxy M61. Optical light curves from the Zwicky Transient Facility, complemented with data from Swift and near-IR photometry are presented. The 350-day duration bolometric light curve exhibits a relatively short (~ 65 days) plateau. This implies a moderate ejecta mass (~ 5 Msun). A series of spectroscopy is presented, including spectropolarimetric observations. The nebular spectra are dominated by Halpha but also reveal emission lines from oxygen and calcium. Comparisons to synthetic nebular spectra indicate an initial progenitor mass of about 12 Msun. Stable nickel is present in the nebular spectrum, with a super-solar Ni/Fe ratio. Several years of pre-discovery data are examined, but no signs of pre-cursor activity is found. Pre-explosion Hubble Space Telescope imaging reveals a probable progenitor star, detected only in the reddest band and is fainter than expected for stars in the 10 - 15 Msun range, in tension with the analysis of the LC and the nebular spectral modeling. We present two additional core-collapse SNe monitored by the ZTF, which also have nebular Halpha-dominated spectra. This illustrates how the absence or presence of interaction with circumstellar material affect both the LCs and in particular the nebular spectra. Type II SN 2020amv has a LC powered by CSM interaction, in particular after about 40 days when the LC is bumpy and slowly evolving. The late-time spectra show strong Halpha emission with a structure suggesting emission from a thin, dense shell. The evolution of the complex three-horn line profile is reminiscent of that observed for SN 1998S. SN 2020jfv has a poorly constrained early-time LC, but shows a transition from a hydrogen-poor Type IIb to a Type IIn, where the nebular spectrum after the light-curve rebrightening is dominated by Halpha, although with an intermediate line width.
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