  High magnetic fields have revealed a surprisingly small Fermi-surface in
underdoped cuprates, possibly resulting from Fermi-surface reconstruction due
to an order parameter that breaks translational symmetry of the crystal
lattice. A crucial issue concerns the doping extent of this state and its
relationship to the principal pseudogap and superconducting phases. We employ
pulsed magnetic field measurements on the cuprate HgBa$_2$CuO$_{4+\delta}$ to
identify signatures of Fermi surface reconstruction from a sign change of the
Hall effect and a peak in the temperature-dependent planar resistivity. We
trace the termination of Fermi-surface reconstruction to two hole
concentrations where the superconducting upper critical fields are found to be
enhanced. One of these points is associated with the pseudogap end-point near
optimal doping. These results connect the Fermi-surface reconstruction to both
superconductivity and the pseudogap phenomena.
