We show by a detailed computational experiment that the three-particle system of protonated oxygen allows us to study the behavior of curve crossings within the Born-Oppenheimer picture in terms of the relative strength of the couplings that exist between the relevant internal coordinates. In particular, we show that the one- and two-parameter seam found in the C-2V symmetry are reduced-dimensionality manifestations of an intersection that actually becomes a conical seam in the three-parameters C-s symmetry. We further show, then, that no trifurcation point exists in this case since the expected C-s seam of the AB(2)-type systems does not appear in the O2H+ species.