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In this paper we consider $C^\infty $-generic families of area-preservingdiffeomorphisms of the torus homotopic to the identity and their rotation sets.Let $f_t:\rm{T^2\rightarrow T^2}$ be such a family, $\widetilde{f}_t:\rmI\negthinspace R^2 \rightarrow \rm I\negthinspace R^2$ be a fixed family oflifts and $\rho (\widetilde{f}_t)$ be their rotation sets, which we assume tohave interior for $t$ in a certain open interval $I.$ We also assume that somerational point $(\frac pq,\frac rq)\in \partial \rho(\widetilde{f}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$ andwe 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 itsrotation 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 thecontinuation of $P_{\overline{t}}$ with the parameter, then$W^u(\widetilde{P}_{t_i})$ (the unstable manifold) has quadratic tangencieswith $W^s(\widetilde{P}_{t_i})+(c,d)$ (the stable manifold translated by$(c,d)),$ where $\widetilde{P}_{t_i}$ is any lift of $P_{t_i}$ to the plane, inother 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 integervectors such that $W^u(\widetilde{P}_{\overline{t}})$ do not intersect$W^s(\widetilde{P}_{\overline{t}})+(c,d).$ And these tangencies becometransverse as $t$ increases.

A consequence of the growth of rotation sets for families of diffeomorphisms of the torus