On completely decomposable defining equations of points in general position in ${\mathbb{P}}^n$

Abstract

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 $\le \lceil \frac{|\Gamma |}{n}\rceil$. 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 $I(\Gamma {)}_{\le \lceil \frac{|\Gamma |}{n}\rceil -1}$ and the set of all completely decomposable forms of degree $\lceil \frac{|\Gamma |}{n}\rceil$ in $I(\Gamma )$. In particular, it holds that if $d\le 2n$ then $I(\Gamma )$ is generated by quadratic equations of rank $2$. This reproves Saint-Donat's results in \cite{SD1} and \cite{SD2}.

References in corpus (1)

• On the variety parametrizing completely decomposable polynomials