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
$f_{t}:\text{T}^{2}\rightarrow \text{T}^{2}$
be such a family,
$\widetilde{f}_{t}:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$
be a fixed family of lifts and
$\unicode[STIX]{x1D70C}(\widetilde{f}_{t})$
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
$(p/q,l/q)\in \unicode[STIX]{x2202}\unicode[STIX]{x1D70C}(\widetilde{f}_{\overline{t}})$
for a certain parameter
$\overline{t}\in I$
, and we want to understand the consequences of the following hypothesis: for all
$t>\overline{t}$
,
$t\in I$
,
$(p/q,l/q)\in \text{int}(\unicode[STIX]{x1D70C}(\widetilde{f}_{t}))$
. 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
$(p/q,l/q)$
. We also prove that 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
$(\widetilde{f}_{t_{i}})^{q}-(p,l)$
, 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. We also prove that, for
$t>\overline{t}$
,
$W^{u}(\widetilde{P}_{t})$
has transverse intersections with
$W^{s}(\widetilde{P}_{t})+(a,b)$
, for all integer vectors
$(a,b)$
, and thus one may consider that the tangencies above are associated to the birth of the heteroclinic intersections in the plane that do not exist for
$t\leq \overline{t}$
.