  The rise and fall of the superconducting transition temperature $T_c$ upon
tuning carrier density or external parameters, such as pressure or magnetic
field, is ubiquitously observed in a wide range of quantum materials. In order
to investigate such domes of $T_c$, we go beyond the prototypical attractive
Hubbard model, and consider a lattice model of electrons coupled via
instantaneous, spatially extended, attractive interactions. By numerically
solving the mean-field equations, as well as going beyond mean field theory
using a functional renormalization group approach, we find that for a
characteristic interaction range $\ell$, there exists a dome in $T_c$ around
$k_F \ell \! \sim \! {\mathcal{O}}(1)$. For multiband systems, our mean field
theory shows the presence of additional domes in the vicinity of Lifshitz
transitions. Our results hold in both two and three dimensions and can be
intuitively understood from the geometric relation between the Fermi surface
and the interaction range. Our model may be relevant for domes of $T_c$ in
dilute weakly coupled superconductors or in engineered cold atom systems.
