  We study spin-1 bosons in an optical lattice under a magnetic field by the
Gutzwiller approximation. Our results thus obtained join the discontinuous
phase boundary curves obtained by perturbative studies through a first-order
transition. On the phase boundary curve, we also find a peculiar cusp structure
originating from the degeneracy between different spin Mott states under a
magnetic field. The magnetic field dependence of both fluctuation in the total
number of bosons and the spin magnetization clarifies that the superfluid phase
is divided into two regions reflecting the coexisting first- and second-order
superfluid transitions.
