  We use computer simulations to study the thermodynamic properties of a glass
former in which a fraction $c$ of the particles has been permanently frozen. By
thermodynamic integration, we determine the Kauzmann, or ideal glass
transition, 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 the
distribution function of the overlap, {\it i.e.}, the order parameter that
signals the glass state. We find that the transition line obtained from the
overlap coincides with that obtained from the thermodynamic integration, thus
showing that the two approaches give the same transition line. Finally we
determine the geometrical properties of the potential energy landscape, notably
the $T-$and $c-$dependence of the saddle index and use these properties to
obtain 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 at
which the glass transition line ends. These findings are qualitatively
consistent with the scenario proposed by the random first order transition
theory.
