A perpendicular magnetic recording data storage system combines a perpendicular medium that has a thin low-magnetic-permeability or "soft" underlayer (SUL) with a recording head that has a trailing shield (TS) with a thick throat height, i.e., a thickness in a direction orthogonal to the recording layer of the medium. The SUL is thin enough and has a low enough magnetic permeability to become saturated in a region beneath the trailing gap of the head during writing, but the throat height of the TS is thick enough to prevent the TS from becoming magnetically saturated during writing. The magnetic saturation of the SUL during writing changes the magnetic reluctance such that more of the magnetic flux going through the SUL changes direction ("field undershoot") and goes to the TS. If the permeability of the SUL is so low (e.g., close to unity) that the SUL does not magnetically saturate, field undershoot may still occur because the reluctance from the SUL to the TS is still smaller than the reluctance from the SUL to the return pole (RP). Field undershoot enables a high write field gradient, which results in narrower magnetic transitions.