  We study the role of the quench temperature $T_f$ in the phase-ordering
kinetics of the Ising model with single spin flip in $d=2,3$. Equilibrium
interfaces are flat at $T_f=0$, whereas at $T_f>0$ they are curved and rough
(above the roughening temperature in $d=3$). We show, by means of scaling
arguments and numerical simulations, that this geometrical difference is
important for the phase-ordering kinetics as well. In particular, while the
growth exponent $z=2$ of the size of domains $L(t)\sim t^{1/z}$ is unaffected
by $T_f$, other exponents related to the interface geometry take different
values at $T_f=0$ or $T_f>0$. For $T_f>0$ a crossover phenomenon is observed
from an early stage where interfaces are still flat and the system behaves as
at $T_f=0$, to the asymptotic regime with curved interfaces characteristic of
$T_f>0$. Furthermore, it is shown that the roughening length, although
sub-dominant with respect to $L(t)$, produces appreciable correction to scaling
up to very long times in $d=2$.
