Cones of Hilbert functions

TL;DR

This paper investigates the geometric structure of cones formed by Hilbert functions of modules over polynomial rings, characterizing their supporting hyperplanes and extreme rays under various boundedness conditions.

Contribution

It provides a detailed description of the convex cones generated by Hilbert functions, including their dimensionality, simpliciality, and polyhedral properties, for different boundedness constraints.

Findings

The cone of all Hilbert functions is infinite-dimensional and simplicial.

The cone of modules with bounded a-invariant is finite-dimensional but not simplicial or polyhedral.

The cone of modules with bounded Castelnuovo-Mumford regularity is finite-dimensional and simplicial.

Abstract

We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded a-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial.