  Superconductivity is often found in a dome around quantum critical points,
i.e. 2nd-order quantum phase transitions. Here, we show that an enhancement of
superconductivity is avoided at the critical pressure of the
charge-density-wave (CDW) state in NbSe$_2$. We present comprehensive
high-pressure Hall effect and magnetic susceptibility measurements of the CDW
and superconducting state in NbSe$_2$. Initially, the 2nd-order CDW transition
is suppressed smoothly but it drops to zero abruptly at PCDW = 4.4 GPa thus
indicating a change to 1st order whilstthe superconducting transition
temperature Tc rises continuously up to PCDW but is constant above. The
putative 1st-order nature of the CDW transition is suggested as the cause for
the absence of a superconducting dome at PCDW. Indeed, we show that the
suppression of the superconducting state at low pressures is due to the loss of
density of states inside the CDW phase whilst the initial suppression of the
CDW state is accounted for by the stiffening of the underlying bare phonon
mode.
