  We investigate the Kovacs (or crossover) effect in facilitated $f$-spin
models of glassy dynamics. Although the Kovacs hump shows a behavior
qualitatively similar for all cases we have examined (irrespective of the
facilitation parameter $f$ and the spatial dimension $d$), we find that the
dependence of the Kovacs peak time on the temperature of the second quench
allows to distinguish among different microscopic mechanisms responsible for
the glassy relaxation (e.g. cooperative vs defect diffusion). We also analyze
the inherent structure dynamics underlying the Kovacs protocol, and find that
the class of facilitated spin models with $d>1$ and $f>1$ shows features
resembling those obtained recently in a realistic model of fragile glass
forming liquid.
