Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial

TL;DR

This paper investigates the minimal Castelnuovo-Mumford regularity for schemes with a given Hilbert polynomial, providing constructive methods and explicit computations, with applications to Hilbert schemes.

Contribution

It introduces new constructive methods based on growth-height-lexicographic Borel sets to compute minimal regularity for schemes with specified Hilbert functions and polynomials.

Findings

Established that minimal regularity can be achieved by schemes with minimal Hilbert functions.

Provided explicit formulas for minimal regularity given Hilbert function regularity.

Developed two new constructive methods: ideal graft and extended lifting.

Abstract

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial $p(z)$. Experimental evidences led us to consider the idea that $m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ of the Hilbert function, and also the minimal Castelnuovo-Mumford regularity $m_u$ of schemes with Hilbert function $u$. These results find applications in the study of Hilbert schemes. They are obtained by means of minimal Hilbert functions 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.