# Mumford's Degree of Contact and Diophantine Approximations

@article{Ferretti2000MumfordsDO,
title={Mumford's Degree of Contact and Diophantine Approximations},
author={Roberto Ferretti},
journal={Compositio Mathematica},
year={2000},
volume={121},
pages={247-262}
}
• R. Ferretti
• Published 2 April 1998
• Mathematics
• Compositio Mathematica
The purpose of this note is to present a somewhat unexpected relation between diophantine approximations and the geometric invariant theory. The link is given by Mumford's degree of contact. We show that destabilizing flags of Chow-unstable projective varieties provide systems of diophantine approximations which are better than those given by Schmidt's subspace theorem, and we give examples of these systems.

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## References

SHOWING 1-10 OF 22 REFERENCES

### Chow instability of certain projective varieties

• S. Ishii
• Mathematics
Nagoya Mathematical Journal
• 1983
A pair (X, D) of a projective variety X and a very ample divisor D on X is called stable (resp. semi-stable, resp. unstable) if the Chow point corresponding to the embedding is SL(N + 1)-stable

### Diophantine Approximations and Value Distribution Theory

Heights and integral points.- Diophantine approximations.- A correspondence with Nevanlinna theory.- Consequences of the main conjecture.- The ramification term.- Approximation to hyperplanes.

### Quantitative diophantine approximations on projective varieties

In this note we study some aspects of a paper of G. Faltings and G. Wüstholz [7]. In particular, we work out a quantitative version of their main theorem. Our result even gives an explicit

### Lectures on Arakelov Geometry

• Mathematics
• 1992
Introduction 1. Intersection theory on regular schemes 2. Green currents 3. Arithmetic Chow groups 4. Characteristic classes 5. The determinant of Laplace operators 6. The determinant of the

### Algebraic Geometry

Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)

### Stability of two-dimensional local ring

• I, Invent. Math
• 1981