  We revisited the phase diagram of the second layer of 4He on top of graphite
using quantum Monte Carlo methods. Our aim was to explore the existence of the
novel phases suggested recently in experimental works, and determine their
properties and stability limits. We found evidence of a superfluid quantum
phase with hexatic correlations, induced by the corrugation of the first Helium
layer, and a quasi-two-dimensional supersolid corresponding to a 7/12
registered phase. The 4/7 commensurate solid was found to be unstable, while
the triangular incommensurate crystals, stable at large densities, were normal.
