  Pinning a fraction of particles from an equilibrium configuration in
supercooled liquids has been recently proposed as a way to induce a new kind of
glass transition, the Random Pinning Glass Transition (RPGT). The RPGT has been
predicted to share some features of standard thermodynamic glass transitions
and usual first order ones. Thanks to its special nature, the approach and the
study of the RPGT appears to be a fairly reachable task compared to the
daunting problem of inspecting standard glass transitions. In this Letter we
generalize the pinning particle procedure. We study a mean-field system where
the pinned configuration is extracted from the equilibrium distribution at
temperature $T'$ and the thermodynamics of the non pinned particles is observed
at a lower temperature $T$. A more complicated physics emerges from this
generalization eventually clarifying the origin and the peculiar
characteristics of the RPGT.
