  We consider a two-dimensional nonlinear Schr\"odinger equation with
concentrated nonlinearity. In both the focusing and defocusing case we prove
local well-posedness, i.e., existence and uniqueness of the solution for short
times, as well as energy and mass conservation. In addition, we prove that this
implies global existence in the defocusing case, irrespective of the power of
the nonlinearity, while in the focusing case blowing-up solutions may arise.
