  We observe that H$_3$S has a bcc structure and, with nearest neighbour
hopping only, a strong singularity occurs at zero energy. This singularity is
accompanied with a highly nested Fermi surface, which is {\it not} conducive to
a stable superconducting instability. Introduction of next-nearest-neighbour
hopping removes the singularity, but a `robust' peak remains in the electron
density of states. Solution of the BCS equations shows an enhanced
superconducting $T_c$ due to this peak. Furthermore, nesting is no longer
present, so other instabilities will not compete effectively with
superconductivity. We find high critical temperatures are possible, even with
very modest coupling strengths. We also examine a limit of the $T=0$ equations
(in an Appendix) where an analytical solution is possible over the entire range
of coupling strengths, and therefore the BCS-BEC crossover is fully covered.
