  Rotation period measurements of stars observed with the Kepler mission have
revealed a lack of stars at intermediate rotation periods, accompanied by a
decrease of photometric variability. Whether this so-called dearth region is a
peculiarity of stars in the Kepler field, or reflects a general manifestation
of stellar magnetic activity, is still under debate. Our goal is to measure
stellar rotation periods and photometric variabilities for tens of thousands of
K2 stars, located in different fields along the ecliptic plane, to shed light
on the relation between stellar rotation and photometric variability. We use
Lomb-Scargle periodograms, auto-correlation and wavelet functions to determine
consistent rotation periods. Stellar brightness variability is assessed by
computing the variability range from the light curve. We further apply Gaussian
mixture models to search for bimodality in the rotation period distribution.
Combining measurements from all K2 campaigns, we detect rotation periods in
29,860 stars. For effective temperatures below 6000K, the variability range
shows a local minimum at different periods, consistent with an isochrone age of
750 Myr. Additionally, the K2 rotation period distribution shows evidence for
bimodality, although the dearth region is less pronounced compared to the
Kepler field. The period at the dip of the bimodal distribution shows good
agreement with the period at the local variability minimum. We conclude that
the period bimodality is present in different fields of the sky, and is hence a
general manifestation of stellar magnetic activity. The reduced variability in
the dearth region is interpreted as a cancelation between dark spots and bright
faculae. Our results strongly advocate that the role of faculae has been
underestimated so far, suggesting a more complex dependence of the brightness
variability on the rotation period.
