On subfiniteness of graded linear series

TL;DR

This paper proves that subfiniteness of graded linear series is independent of the ambient function field, extending Hilbert's 14th problem to a graded setting and applying it to geometric and arithmetic series.

Contribution

It establishes a graded analogue of the subfiniteness property, showing it does not depend on the function field, which advances understanding of graded linear series.

Findings

Subfiniteness of graded linear series is independent of the function field.

A graded analogue of Hilbert's 14th problem is established.

Applications to geometric and arithmetic graded linear series are demonstrated.

Abstract

Hilbert's 14th problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the field of fractions of the algebra. It has a negative answer due to the counterexample of Nagata. We show that a subfinite version of Hilbert's 14th problem has a confirmative answer. We then establish a graded analogue of this result, which permits to show that the subfiniteness of graded linear series does not depend on the function field in which we consider it. Finally, we apply the subfiniteness result to the study of geometric and arithmetic graded linear series.