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We use computer simulations to study the thermodynamic properties of a glassformer in which a fraction $c$ of the particles has been permanently frozen. Bythermodynamic integration, we determine the Kauzmann, or ideal glasstransition, temperature $T_K(c)$ at which the configurational entropy vanishes.This is done without resorting to any kind of extrapolation, {\it i.e.},$T_K(c)$ is indeed an equilibrium property of the system. We also measure thedistribution function of the overlap, {\it i.e.}, the order parameter thatsignals the glass state. We find that the transition line obtained from theoverlap coincides with that obtained from the thermodynamic integration, thusshowing that the two approaches give the same transition line. Finally wedetermine the geometrical properties of the potential energy landscape, notablythe $T-$and $c-$dependence of the saddle index and use these properties toobtain the dynamic transition temperature $T_d(c)$. The two temperatures$T_K(c)$ and $T_d(c)$ cross at a finite value of $c$ and indicate the point atwhich the glass transition line ends. These findings are qualitativelyconsistent with the scenario proposed by the random first order transitiontheory.

Equilibrium phase diagram of a randomly pinned glass-former