  In this work we investigate positivity properties of nonlocal Schr\"odinger
type operators, driven by the fractional Laplacian, with multipolar, critical,
and locally homogeneous potentials. On one hand, we develop a criterion that
links the positivity of the spectrum of such operators with the existence of
certain positive supersolutions, while, on the other hand, we study the
localization of binding for this kind of potentials. Combining these two tools
and performing an inductive procedure on the number of poles, we establish
necessary and sufficient conditions for the existence of a configuration of
poles that ensures the positivity of the corresponding Schr\"odinger operator.
