  Four-dimensional state space geometry is worked out for the exactly solved
one-dimensional spin-3/2 lattice with a Blume-Emery-Griffiths (BEG) Hamiltonian
as well as a more general one with a term containing a non-zero field coupling
to the octopole moments. The phase behaviour of the spin-3/2 chain is also
explored extensively and novel phenomena suggesting anomalies in the
hyperscaling relation and in the decay of fluctuations are reported for a range
of parameter values. Using the method of constrained fluctuations worked out
earlier in \cite{asknbads,riekan1} three sectional curvatures and a $3d$
curvature are obtained and shown to separately encode dipolar, quadrupolar and
octopolar correlations both near and away from pseudo-criticality. In all
instances of a seeming hyperscaling violation the $3d$ scalar curvature is
found to encode the correlation length while the relevant $2d$ curvature equals
the inverse of singular free energy. For parameter values where the order
parameter fluctuation anomalously decays despite a divergence in correlation
length the relevant scalar curvature undergoes a sign change to positive
values, signalling a possible change in statistics.
