  Dirac semimetals (DSMs) are an important class of topological states of
matter. Here, focusing on DSMs of band inversion type, we investigate their
boundary modes from the effective model perspective. We show that in order to
properly capture the boundary modes, $k$-cubic terms must be included in the
effective model, which would drive an evolution of surface degeneracy manifold
from a nodal line to a nodal point. Using first-principles calculations, we
demonstrate that this feature and the topological hinge modes can be clearly
exhibited in $\beta$-CuI. We further extend the discussion to magnetic DSMs and
show that the time-reversal symmetry breaking can gap out the surface bands and
hence help to expose the hinge modes in the spectrum, which could be beneficial
for the experimental detection of hinge modes.
