  In this paper we propose a short range generalization of the $p$-spin
interaction spin-glass model. The model is well suited to test the idea that an
entropy collapse is at the bottom-line of the dynamical singularity encountered
in structural glasses. The model is studied in three dimensions through Monte
Carlo simulations, which put in evidence fragile glass behavior with stretched
exponential relaxation and super-Arrhenius behavior of the relaxation time. Our
data are in favor of a Vogel-Fulcher behavior of the relaxation time, related
to an entropy collapse at the Kauzmann temperature. We however encounter
difficulties analogous to those found in experimental systems when
extrapolating thermodynamical data at low temperatures. We study the spin glass
susceptibility investigating the behavior of the correlation length in the
system. We find that the the increase of the relaxation time is not accompanied
by any growth of the correlation length. We discuss the scaling properties of
off-equilibrium dynamics in the glassy regime, finding qualitative agreement
with the mean-field theory.
