On subfiniteness of graded linear series

Abstract

Hilbert’s fourteenth problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the fraction field of the algebra. It has a negative answer due to a counterexample of Nagata. We show that a subfinite version of Hilbert’s fourteenth problem has an affirmative 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.