  In this paper we consider $C^\infty $-generic families of area-preserving
diffeomorphisms of the torus homotopic to the identity and their rotation sets.
Let $f_t:\rm{T^2\rightarrow T^2}$ be such a family, $\widetilde{f}_t:\rm
I\negthinspace R^2 \rightarrow \rm I\negthinspace R^2$ be a fixed family of
lifts and $\rho (\widetilde{f}_t)$ be their rotation sets, which we assume to
have interior for $t$ in a certain open interval $I.$ We also assume that some
rational point $(\frac pq,\frac rq)\in \partial \rho
(\widetilde{f}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$ and
we want to understand consequences of the following hypothesis: For all
$t>\overline{t},$ $t\in I,$ $(\frac pq,\frac rq)\in int(\partial \rho
(\widetilde{f}_t)).$
  Under these very natural assumptions, we prove that there exists a
$f_{\overline{t}}^q$-fixed hyperbolic saddle $P_{\overline{t}}$ such that its
rotation vector is $(\frac pq,\frac rq)$ and, there exists a sequence
$t_i>\overline{t},$ $t_i\rightarrow \overline{t},$ such that if $P_t$ is the
continuation of $P_{\overline{t}}$ with the parameter, then
$W^u(\widetilde{P}_{t_i})$ (the unstable manifold) has quadratic tangencies
with $W^s(\widetilde{P}_{t_i})+(c,d)$ (the stable manifold translated by
$(c,d)),$ where $\widetilde{P}_{t_i}$ is any lift of $P_{t_i}$ to the plane, in
other words, $\widetilde{P}_{t_i}$ is a fixed point for
$(\widetilde{f}_{t_i})^q-(p,r),$ and $(c,d)\neq (0,0)$ are certain integer
vectors such that $W^u(\widetilde{P}_{\overline{t}})$ do not intersect
$W^s(\widetilde{P}_{\overline{t}})+(c,d).$ And these tangencies become
transverse as $t$ increases.
