  The properties of cuprate high-temperature superconductors are largely shaped
by competing phases whose nature is often a mystery. Chiefly among them is the
pseudogap phase, which sets in at a doping $p^*$ that is material-dependent.
What determines $p^*$ is currently an open question. Here we show that the
pseudogap cannot open on an electron-like Fermi surface, and can only exist
below the doping $p_{FS}$ at which the large Fermi surface goes from hole-like
to electron-like, so that $p^*$ $\leq$ $p_{FS}$. We derive this result from
high-magnetic-field transport measurements in
La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ under pressure, which reveal a large and
unexpected shift of $p^*$ with pressure, driven by a corresponding shift in
$p_{FS}$. This necessary condition for pseudogap formation, imposed by details
of the Fermi surface, is a strong constraint for theories of the pseudogap
phase. Our finding that $p^*$ can be tuned with a modest pressure opens a new
route for experimental studies of the pseudogap.
