  Motivated by a kind of Penrose correspondence, we investigate the space of
hyperplane sections of Segre quartic surfaces which have an ordinary cusp. We
show that the space of such hyperplane sections is empty for two kinds of Segre
surfaces, and it is a connected surface for all other kinds of Segre surfaces.
We also show that when it is non-empty, the closure of the space is either
birational to the surface itself or birational to a double covering of the
surface, whose branch divisor consists of some specific lines on the surface.
