The study of the defining equations of a finite set $\Gamma \subset \mathbb{ P}^n$ in linearly general position has been actively attracted since it plays a significant role in understanding the defining equations of arithmetically Cohen-Macaulay varieties. In \cite{ T}, R. Treger proved that $I(\Gamma)$ is generated by forms of degree $\leq \lceil \frac{ |\Gamma|}{ n}\rceil$. Since then, Treger's result have been extended and improved in several papers. The aim of this paper is to reprove and improve the above Treger's result from a new perspective. Our main result in this paper shows that $I(\Gamma)$ is generated by the union of $I(\Gamma)_{ \leq \lceil \frac{ |\Gamma|}{ n}\rceil -1}$ and the set of all completely decomposable forms of degree $\lceil \frac{ |\Gamma|}{ n}\rceil$ in $I(\Gamma)$. In particular, it holds that if $d \leq 2n$ then $I(\Gamma)$ is generated by quadratic equations of rank $2$. This reproves Saint-Donat's results in \cite{ SD1} and \cite{ SD2}.
Categories: math.AG math.AC