  We study the quadratic Zeeman effect (QZE) in a system of antiferromagnetic
spin-1 bosons on a square lattice and derive the ground-state phase diagrams by
means of quantum Monte Carlo simulations and mean field treatment. The QZE
imbalances the populations of the magnetic sublevels $\sigma=\pm1$ and
$\sigma=0$, and therefore affects the magnetic and mobility properties of the
phases. Both methods show that the tip of the even Mott lobes, stabilized by
singlet state, is destroyed when turning on the QZE, thus leaving the space to
the superfluid phase. Contrariwise, the tips of odd Mott lobes remain
unaffected. Therefore, the Mott-superfluid transition with even filling
strongly depends on the strength of the QZE, and we show that the QZE can act
as a control parameter for this transition at fixed hopping. Using quantum
Monte Carlo simulations, we elucidate the nature of the phase transitions and
examine in detail the nematic order: the first-order Mott-superfluid transition
with even filling observed in the absence of QZE becomes second order for weak
QZE, in contradistinction to our mean-field results which predict a first-order
transition in a larger range of QZE. Furthermore, a spin nematic order with
director along the $z$ axis is found in the odd Mott lobes and in the
superfluid phase for energetically favored $\sigma=\pm1$ states. In the
superfluid phase with even filling, the $xy$ components of the nematic director
remain finite only for moderate QZE.
