Mumford's Degree of Contact and Diophantine Approximations

Abstract

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 $s>0$.