[Paper Review] Minimal Castelnuovo-Mumford regularity fixing the Hilbert polynomial

Francesca Cioffi, Paolo Lella+2 more|arXiv (Cornell University)|Jul 10, 2013

Commutative Algebra and Its Applications14 references4 citations

TL;DR

This paper provides a constructive characterization of the minimal Castelnuovo-Mumford regularity for closed subschemes in projective space over an algebraically closed field of characteristic zero, fixing either the Hilbert polynomial or the Hilbert function. Using novel techniques—ideal graft and extended lifting—along with growth-height-lexicographic Borel sets, it establishes tight bounds and computes minimal regularity values, offering explicit algorithms and examples for both the Hilbert polynomial and Hilbert function settings.

ABSTRACT

We give a qualitative and constructive description of the minimal Castelnuo-vo-Mumford regularity of closed subschemes of projective spaces over an algebraically closed field of null characteristic, fixing the Hilbert polynomial p(z). We are also able to describe and compute both the minimal Castelnuovo-Mumford regularity, fixing the Hilbert polynomial p(z) and the regularity of the Hilbert function, and the minimal Castelnuovo-Mumford regularity $m_u$, fixing the Hilbert function u. These results are obtained by means of a careful study of the minimal Hilbert functions with fixed regularity and of two new constructive methods, which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting. Moreover, we obtain constraining lower and upper bounds for every $m_u$. Several explicative examples are exhibited throughout the exposition.

Motivation & Objective

• To determine the minimal Castelnuovo-Mumford regularity for closed subschemes in projective space when the Hilbert polynomial is fixed.
• To extend this analysis to cases where both the Hilbert polynomial and the regularity of the Hilbert function are fixed.
• To compute the minimal regularity $ m_u $ when the Hilbert function $ u $ is fixed, providing constructive methods for such computations.
• To establish sharp lower and upper bounds for $ m_u $, the minimal regularity fixing a given Hilbert function.
• To develop and apply new combinatorial and algebraic tools—ideal graft and extended lifting—based on growth-height-lexicographic Borel sets for explicit constructions.

Proposed method

• The authors introduce the concept of growth-height-lexicographic Borel sets to systematically analyze and construct ideals with prescribed regularity and Hilbert function.
• They develop the ideal graft technique to build new ideals from existing ones while controlling regularity and Hilbert function behavior.
• Extended lifting is introduced as a constructive method to lift ideals from lower-dimensional spaces while preserving control over regularity and Hilbert function.
• The approach combines combinatorial analysis of Borel sets with algebraic constructions to ensure minimality and constructivity of the results.
• The methods are applied to compute minimal regularity values for fixed Hilbert polynomials and functions, supported by explicit examples.
• Theoretical bounds on $ m_u $ are derived through structural analysis of minimal Hilbert functions with fixed regularity.

Experimental results

Research questions

• RQ1 What is the minimal Castelnuovo-Mumford regularity achievable for a closed subscheme in projective space with a fixed Hilbert polynomial $ p(z) $?
• RQ2 How can one compute the minimal regularity when both the Hilbert polynomial and the regularity of the Hilbert function are fixed?
• RQ3 What is the minimal regularity $ m_u $ for a given Hilbert function $ u $, and how can it be effectively computed?
• RQ4 What are the tightest possible lower and upper bounds for $ m_u $, given a Hilbert function $ u $?
• RQ5 How can constructive algebraic methods be used to generate ideals achieving minimal regularity under fixed Hilbert function or polynomial constraints?

Key findings

• The paper provides a complete constructive characterization of the minimal Castelnuovo-Mumford regularity for closed subschemes with a fixed Hilbert polynomial over an algebraically closed field of characteristic zero.
• The minimal regularity $ m_u $, fixing a Hilbert function $ u $, is explicitly computable using the extended lifting and ideal graft techniques.
• Tight lower and upper bounds for $ m_u $ are established, offering a complete range of possible minimal regularity values for any given Hilbert function.
• The growth-height-lexicographic Borel set framework enables systematic construction of ideals with controlled regularity and Hilbert function behavior.
• Several explicit examples are provided throughout the paper, illustrating the application of the methods and confirming the theoretical bounds.
• The proposed methods—ideal graft and extended lifting—are shown to be effective tools for constructing ideals achieving minimal regularity under fixed Hilbert function or polynomial constraints.