Regularity and Multisecant Lines of Finite Schemes

Abstract For a nondegenerate finite subscheme $\Gamma$ in ${\mathbb P}^c$, let ${\rm reg}(\Gamma)$ and $\ell (\Gamma)$ be, respectively, the regularity of $\Gamma$ and the largest integer $\ell$ such that there exists an $\ell$-secant line to $\Gamma$. It is always true that ${\rm reg}(\Gamma) \geq \ell (\Gamma)$. In this article, we show that if ${\rm reg}(\Gamma) \geq \frac{d-c+5}{2}$ then ${\rm reg}(\Gamma)$ is equal to $\ell (\Gamma)$. In addition, we describe the minimal free resolution of the homogeneous ideal of $\Gamma$ for the case ${\rm reg}(\Gamma) \geq \frac{d-c+5}{2}$.