  We study quantum phase transitions in the ground state of the two dimensional
hard-core boson Hubbard Hamiltonian. Recent work on this and related models has
suggested ``supersolid'' phases with simultaneous diagonal and off-diagonal
long range order. We show numerically that, contrary to the generally held
belief, the most commonly discussed ``checkerboard'' supersolid is
thermodynamically unstable. Furthermore, this supersolid cannot be stabilized
by next near neighbour interaction. We obtain the correct phase diagram using
the Maxwell construction. We demonstrate the ``striped'' supersolid is
thermodynamically stable and is separated from the superfluid phase by a
continuous phase transition.
