1998/04/02 by Roberto Ferretti, Roberto G. Ferretti, Ferretti, Roberto G.
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Topological and Geometric Data Analysis #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.math/9804011
12 pages, LaTex2e
arxiv created 1998/04/02 · openalex publication_date 1998/04/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Pn, does there exists a system of diophantine approximations on Pn whose solutions are Zariski dense in Pn, but lie in finitely many proper subvarieties of X? One can gain insight into this problem using a theorem of G. Faltings and G. Wüstholz. 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-Wüstholz 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 Xs, for some large s>0.