  A deep conjecture on torsion anomalous varieties states that if $V$ is a
weak-transverse variety in an abelian variety, then the complement $V^{ta}$ of
all $V$-torsion anomalous varieties is open and dense in $V$. We prove some
cases of this conjecture. We show that the $V$-torsion anomalous varieties of
relative codimension one are non-dense in any weak-transverse variety $V$
embedded in a product of elliptic curves with CM. We give explicit uniform
bounds in the dependence on $V$. As an immediate consequence we prove the
conjecture for $V$ of codimension two in a product of CM elliptic curves. We
also point out some implications on the effective Mordell-Lang Conjecture.
