Castelnuovo-Mumford regularity of finite schemes

TL;DR

This paper establishes an upper bound for the Castelnuovo-Mumford regularity of finite schemes in projective space and characterizes when this bound is nearly attained through the existence of a unique rational normal curve.

Contribution

The paper provides a new upper bound for the regularity of finite schemes and characterizes the schemes that nearly attain this bound via a unique rational normal curve.

Findings

Upper bound for regularity based on degree and secant plane

Characterization of schemes close to the bound via a rational normal curve

Existence and uniqueness condition for the rational normal curve

Abstract

Let $\Gamma \subset \mathbb{P}^n$ be a nondegenerate finite subscheme of degree $d$. Then the Castelnuovo-Mumford regularity ${\rm reg} ({\Gamma})$ of $\Gamma$ is at most $\left\lceil \frac{d-n-1}{t(\Gamma)} \right\rceil +2$ where $t(\Gamma)$ is the smallest integer such that $\Gamma$ admits a $(t+2)$-secant $t$-plane. In this paper, we show that ${\rm reg} ({\Gamma})$ is close to this upper bound if and only if there exists a unique rational normal curve $C$ of degree $t(\Gamma)$ such that ${\rm reg} (\Gamma \cap C) = {\rm reg} (\Gamma)$.