  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}.
