  We study the existence, the nonexistence, and the shape of the ground states
of a Nonlinear Schr\"odinger Equation on a manifold called hybrid plane, that
consists of a half-line whose origin is connected to a plane. The nonlinearity
is of power type, focusing and subcritical. The energy is the sum of the
Nonlinear Schr\"odinger energies with a contact interaction on the half-line
and on the plane with an additional quadratic term that couples the two
components. By ground state we mean every minimizer of the energy at a fixed
mass.
  As a first result, we single out the following rule: a ground state exists if
and only if the confinement near the junction is energetically more convenient
than escaping at infinity along the halfline, while escaping through the plane
is shown to be never convenient. The problem of existence reduces then to a
competition with the one-dimensional solitons.
  By this criterion, we prove existence of ground states for large and small
values of the mass. Moreover, we show that at given mass a ground state exists
if one of the following conditions is satisfied: the interaction at the origin
of the half-line is not too repulsive; the interaction at the origin of the
plane is sufficiently attractive; the coupling between the half-line and the
plane is strong enough. On the other hand, nonexistence holds if the contact
interactions on the half-line and on the plane are repulsive enough and the
coupling is not too strong.
  Finally, we provide qualitative features of ground states. In particular, we
show that in the presence of coupling every ground state is supported both on
the half-line and on the plane and each component has the shape of a ground
state at its mass for the related Nonlinear Schr\"odinger energy with a
suitable contact interaction.
  These are the first results for the Nonlinear Schr\"odinger Equation on a
manifold of mixed dimensionality.
