  In the family of the iron-based superconductors, the $RE$FeAsO-type compounds
(with $RE$ being a rare-earth metal) exhibit the highest bulk superconducting
transition temperatures ($T_{\mathrm{c}}$) up to $55\ \textrm{K}$ and thus hold
the key to the elusive pairing mechanism. Recently, it has been demonstrated
that the intrinsic electronic structure of SmFe$_{0.92}$Co$_{0.08}$AsO
($T_{\mathrm{c}}=18\ \textrm{K}$) is highly nontrivial and consists of multiple
band-edge singularities in close proximity to the Fermi level. However, it
remains unclear whether these singularities are generic to the $RE$FeAsO-type
materials and if so, whether their exact topology is responsible for the
aforementioned record $T_{\mathrm{c}}$. In this work, we use angle-resolved
photoemission spectroscopy (ARPES) to investigate the inherent electronic
structure of the NdFeAsO$_{0.6}$F$_{0.4}$ compound with a twice higher
$T_{\mathrm{c}}=38\ \textrm{K}$. We find a similarly singular Fermi surface and
further demonstrate that the dramatic enhancement of superconductivity in this
compound correlates closely with the fine-tuning of one of the band-edge
singularities to within a fraction of the superconducting energy gap $\Delta$
below the Fermi level. Our results provide compelling evidence that the
band-structure singularities near the Fermi level in the iron-based
superconductors must be explicitly accounted for in any attempt to understand
the mechanism of superconducting pairing in these materials.
