  In the presence of certain symmetries, three-dimensional Dirac semimetals can
harbor not only surface Fermi arcs, but also surface Dirac cones. Motivated by
the experimental observation of rotation-symmetry-protected Dirac semimetal
states in iron-based superconductors, we investigate the potential intrinsic
topological phases in a $C_{4z}$-rotational invariant superconducting Dirac
semimetal with $s_{\pm}$-wave pairing. When the normal state harbors only
surface Fermi arcs on the side surfaces, we find that an interesting gapped
superconducting state with a quartet of Bogoliubov-Dirac cones on each side
surface can be realized, even though the first-order topology of its bulk is
trivial. When the normal state simultaneously harbors surface Fermi arcs and
surface Dirac cones, we find that a second-order time-reversal invariant
topological superconductor with helical Majorana hinge states can be realized.
The criteria for these two distinct topological phases have a simple geometric
interpretation in terms of three characteristic surfaces in momentum space. By
reducing the bulk material to a thin film normal to the axis of rotation
symmetry, we further find that a two-dimensional first-order time-reversal
invariant topological superconductor can be realized if the inversion symmetry
is broken by applying a gate voltage. Our work reveals that diverse topological
superconducting phases and types of Majorana modes can be realized in
superconducting Dirac semimetals.
