Castelnuovo-Mumford regularity of finite schemes

Abstract

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

1 page, LaTeX; typos in the abstract corrected