Diophantine approximation constants for varieties over function fields

TL;DR

This paper extends Diophantine approximation concepts to varieties over function fields, using advanced theorems to relate approximation constants to geometric invariants like volume and Seshadri constants.

Contribution

It introduces a framework for approximation constants over function fields and links them to geometric measures, utilizing an effective Schmidt's subspace theorem.

Findings

Approximation constants can be computed on proper subvarieties under certain conditions.

Approximation constants relate to volume functions and Seshadri constants.

The approach generalizes Diophantine approximation to higher-dimensional varieties.

Abstract

By analogy with the program of McKinnon-Roth, we define and study approximation constants for points of a projective variety X defined over K the function field of an irreducible and non-singular in codimension 1 projective variety defined over an algebraically closed field of characteristic zero. In this setting, we use an effective version of Schmidt's subspace theorem, due to J.T.-Y. Wang, to give a sufficient condition for such approximation constants to be computed on a proper K-subvariety of X. We also indicate how our approximation constants are related to volume functions and Seshadri constants.