Mumford's Degree of Contact and Diophantine Approximations

摘要

The Schmidt Subspace Theorem establishes that the solutions of some particularsystems of diophantine approximations in projective spaces accumulates on a finite number of linear subspaces (10, Theorem V.1.D'). One may state the following questions: given a subvariety X of a projective space Pn, does there exists a system of diophantine approximations on Pn whose solutions in Pn are Zariski dens (in Pn) but whose solutions in X lie in finitely many proper subvarieties of X. One can gain insight into this problem using a theorem of G. Faltings and G. Wustholz (2), Theorem 7.3, or (1) for a quantitative version. 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.