  We translate the problem of designing a secure communications protocol for
several users communicating through a relay in a wireless network into
understanding certain subvarieties of products of Grassmannians. We calculate
the dimension of these subvarieties and provide various results concerning
their defning equations. When the relay and all of the users have the same
number of antennas, this approach places fundamental limits on the amount of
data that can be passed through such a network.
