Planetary Systems around Low-mass Stars Unveiled by K2

We present the detection and follow-up observations of planetary candidates around low-mass stars observed by the {it K2} mission. Based on light-curve analysis, adaptive-optics imaging, and optical spectroscopy at low and high resolution (including radial velocity measurements), we validate 16 planets around 12 low-mass stars observed during {it K2} campaigns 5--10. Among the 16 planets, 12 are newly validated, with orbital periods ranging from 0.96--33 days. For one of the planets (EPIC 220621087.01) we present ground-based transit photometry, allowing us to refine the ephemerides. We also identify EPIC 220187552 as a false positive, based on the multiple stars seen in a high-resolution image and double lines in a high-resolution spectrum. Combining our {it K2} M-dwarf planets together with the validated or confirmed planets found previously, we investigate the dependence of planet radius $R_p$ on stellar insolation and metallicity [Fe/H]. We confirm that medium-sized planets ($R_p=2-5~R_oplus$) seem to have experienced shrinkage --- plausibly due to photoevaporation --- and we find evidence that the shrinkage occurs at lower insolation for the coolest M dwarfs. Planets larger than $approx3~R_oplus$ are only found around the most metal-rich M dwarfs, and for the coolest M dwarfs ($lesssim3500$ K) there appears to be a correlation between planet size and metallicity.