  The Schmidt Subspace Theorem affirms that the solutions of some particular
system of diophantine approximations in projective spaces accumulates on a
finite number of proper linear subspaces. Given a subvariety $X$ of a
projective space $P^n$, does there exists a system of diophantine
approximations on $P^n$ whose solutions are Zariski dense in $P^n$, but lie in
finitely many proper subvarieties of $X$? One can gain insight into this
problem using a theorem of G. Faltings and G. W\"ustholz. Their construction
requires the hypothesis that the sum of some expected values has to be large.
This sum turns out to be proportional to a degree of contact of a weighted flag
of sections over the variety $X$. This invariant measures the semistability of
the Chow (or Hilbert) point of $X$ under the action of an appropriate reductive
algebraic group. Whence, the lower bound in the Faltings-W\"ustholz theorem may
be translated into a GIT language. This means that in order to show that a
system of diophantine approximations on $X$ is not under the control of Schmidt
Subspace Theorem, we must check the Chow-unstability of $X^s$, for some large
$s>0$.
