Approximable algebras and a question of Chen

TL;DR

This paper investigates the concept of approximable graded algebras introduced by Huayi Chen and provides a counter-example demonstrating that not all such algebras are subalgebras of graded section rings of big line bundles.

Contribution

The paper presents a counter-example to Chen's question, showing that some approximable graded algebras cannot be embedded into graded section rings of big line bundles.

Findings

Counter-example to Chen's question

Not all approximable algebras are subalgebras of big line bundle section rings

Clarifies limitations of approximable algebra embeddings

Abstract

In his article "Arithmetic Fujita Approximations", Huayi Chen introduces the notion of an approximable graded algebra and asks if any such algebra is a subalgebra of the graded section ring of a big line bundle on an algebraic variety. We give a counter-example showing this is not the case.