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

TL;DR

This paper improves understanding of the generators of defining ideals of points in general position in projective space, showing they are generated by specific decomposable forms and quadratic equations of rank 2.

Contribution

It provides a new proof and enhancement of Treger's result, demonstrating that the ideal is generated by decomposable forms and lower-degree forms, extending classical results.

Findings

Ideal generated by decomposable forms of degree $ ceil | ext{points}|/n ceil$

If degree $d extless= 2n$, ideal generated by quadratic rank 2 equations

Reproves and extends Saint-Donat's classical results

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 $\leq \lceil \frac{|\Gamma|}{n}\rceil$. 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}.