  We study the surface behavior of the two-dimensional columnar dimerized
quantum antiferromagnetic XXZ model with easy-plane anisotropy, with particular
emphasis on the surface critical behaviors of the (2+1)-dimensional quantum
critical points of the model that belong to the classical three-dimensional
O(2) universality class, for both $S=1/2$ and $S=1$ spins using quantum Monte
Carlo simulations. We find completely different surface behaviors on two
different surfaces of geometrical settings: the dangling-ladder surface, which
is exposed by cutting a row of weak bonds, and the dangling-chain surface,
which is formed by cutting a row of strong bonds along the direction
perpendicular to the strong bonds of a periodic system. Similar to the
Heisenberg limit, we find an ordinary transition on the dangling-ladder surface
for both $S=1$ and $S=1/2$ spin systems. However, the dangling-chain surface
shows much richer surface behaviors than in the Heisenberg limit. For the
$S=1/2$ easy-plane model, at the bulk critical point, we provide evidence
supporting an extraordinary surface transition with a long-range order
established by effective long-range interactions due to bulk critical
fluctuations. The possibility that the state is an extraordinary-log state
seems unlikely. For the $S=1$ system, we find surface behaviors similar to that
of the three-dimensional classical XY model with sufficiently enhanced surface
coupling, suggesting an extraordinary-log state at the bulk critical point.
