  In the framework of the Ginzburg-Landau equation, the temperature dependence
of the upper critical field of small ring-like superconductors is studied. At
equilibrium small parts of the phase diagram show paramagnetism for width /
radius ratios below 0.85. Their number and extension increase with the size of
the hole. In these regions, only the inner part of the ring shows a positive
magnetic moment. The order parameter density profile appears to change, when
crossing a first order transition line, which separates different angular
momentum values, and we clarify the relationship between the localization of
superconductivity nucleation and paramagnetism of those samples.
