Complete intersections with given Hilbert polynomials

TL;DR

This paper investigates when the degrees of generators of a complete intersection are uniquely determined by its Hilbert polynomial, providing criteria and a complete classification up to codimension 6.

Contribution

It offers new criteria for when the generator degrees are determined by the Hilbert polynomial and fully classifies these cases for codimension up to 6.

Findings

Criteria for determining generator degrees from Hilbert polynomial

Complete classification of such cases up to codimension 6

Identification of cases where degrees are not determined by the Hilbert polynomial

Abstract

The Hilbert polynomial of a homogeneous complete intersection is determined by the degrees of the generators of the defining ideal. The degrees of the generators are not, in general, determined by the Hilbert polynomial -- but sometimes they are. When? We give some general criteria and completely answer the question up to codimension 6.