Castelnuovo-Mumford regularity of finite schemes

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 ${\rm reg} (\Gamma \cap C) = {\rm reg} (\Gamma)$.