The Gotzmann Coefficients of Hilbert Functions

TL;DR

This paper explores algebraic and geometric implications of extremal bounds on Hilbert functions, improving existing results and providing new restrictions on Hilbert functions of certain algebraic schemes.

Contribution

It extends Green's Hyperplane Restriction Theorem and offers new conditions for Hilbert polynomials and functions of algebraic schemes.

Findings

Improved bounds on Hilbert functions of hyperplane sections.

New restrictions on Hilbert functions of zero-dimensional schemes.

Conditions for polynomials to be Hilbert polynomials of certain varieties.

Abstract

In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric consequences improve some results in this direction first given by Green and extend others by Bigatti, Geramita, and Migliore. Other applications of our detailed investigation of how the Hilbert polynomial is written as a sum of binomials, are to conditions that must be satisfied by a polynomial if it is to be the Hilbert polynomial of a non-degenerate integral subscheme of $\mathbb P^n$ (a problem posed by R. Stanley). We also give some new restrictions on the Hilbert function of a zero dimensional reduced scheme with the Uniform Position Property.