Castelnuovo-Mumford Regularity of Finite Schemes

초록

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

키워드