  In several unconventional superconductors, the highest superconducting
transition temperature $T_{c}$ is found in a region of the phase diagram where
the antiferromagnetic transition temperature extrapolates to zero, signaling a
putative quantum critical point. The elucidation of the interplay between these
two phenomena - high-$T_{c}$ superconductivity and magnetic quantum criticality
- remains an important piece of the complex puzzle of unconventional
superconductivity. In this paper, we combine sign-problem-free Quantum Monte
Carlo simulations and field-theoretical analytical calculations to unveil the
microscopic mechanism responsible for the superconducting instability of a
general low-energy model, called spin-fermion model. In this approach,
low-energy electronic states interact with each other via the exchange of
quantum critical magnetic fluctuations. We find that even in the regime of
moderately strong interactions, both the superconducting transition temperature
and the pairing susceptibility are governed not by the properties of the entire
Fermi surface, but instead by the properties of small portions of the Fermi
surface called hot spots. Moreover, $T_{c}$ increases with increasing
interaction strength, until it starts to saturate at the crossover from
hot-spots dominated to Fermi-surface dominated pairing. Our work provides not
only invaluable insights into the system parameters that most strongly affect
$T_{c}$, but also important benchmarks to assess the origin of
superconductivity in both microscopic models and actual materials.
