A generalized Schmidt subspace theorem for closed subschemes

TL;DR

This paper extends Schmidt's subspace theorem to closed subschemes using Seshadri constants, providing a unified geometric framework and a shorter proof for higher-dimensional Diophantine approximation results.

Contribution

It introduces a generalized subspace theorem for closed subschemes utilizing Seshadri constants, broadening the scope of Diophantine approximation techniques.

Findings

Unified geometric framework for subspace theorem

Shortened proof of Roth-type approximation theorem

Extended application of Seshadri constants in Diophantine geometry

Abstract

We prove a generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of suitably defined Seshadri constants with respect to a fixed ample divisor. Our proof builds on previous work by Evertse and Ferretti, Corvaja and Zannier, and others, and uses standard techniques from algebraic geometry such as notions of positivity, blowing-ups and direct image sheaves. As an application, we recover a higher-dimensional Diophantine approximation theorem of K.F. Roth-type due to D. McKinnon and M. Roth with a significantly shortened proof, while simultaneously extending the scope of the use of Seshadri constants in this context in a natural way.