Let \(S=k[x_0,x_1,\ldots,x_n]\) be the standard graded polynomial ring over a field \(k\). The purpose of this paper is to describe the cones of Hilbert functions for some classes of \(S\)-modules. For some obvious reason, the \(S\)-modules considered in this paper are all assumed to be finitely generated in degree \(0\).
Mainly, there are three classes of \(S\)-modules considered in this paper: first, the class of all Artinian \(S\)-modules; second, the class of all \(S\)-modules with bounded a-invariant; third, the class of all \(S\)-modules with bounded Castelnuovo-Mumford regularity.
To describe the cone of Hilbert functions of each class of \(S\)-modules, the authors obtained the equations of the supporting hyperplanes of the cone and the \(S\)-modules which generate the extreme rays of the cone. The first cone is infinite dimensional and the other two are finite dimensional.