  We investigate the ground states for the focusing, subcritical nonlinear
Schr\"odinger equation with a point defect in dimension two, defined as the
minimizers of the energy functional at fixed mass. We prove that ground states
exist for every positive mass and show a logarithmic singularity at the defect.
Moreover, up to a multiplication by a constant phase, they are positive,
radially symmetric, and decreasing along the radial direction. In order to
overcome the obstacles arising from the uncommon structure of the energy space,
that complicates the application of standard rearrangement theory, we move to
the study of the minimizers of the action functional on the Nehari manifold and
then establish a connection with the original problem. An ad hoc result on
rearrangements is given to prove qualitative features of the ground states.
