Castelnuovo-Mumford regularity of finite schemes

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