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

TL;DR

Let $I(\Gamma)$ denote the homogeneous ideal of a finite set $\Gamma \subset \mathbb{P}^n$ in linearly general position. The paper shows that, when $d=|\Gamma| \le mn$, $I(\Gamma)$ is generated by $I(\Gamma)_{\le m-1}$ together with all completely decomposable degree-$m$ forms in $I(\Gamma)$, i.e. $I(\Gamma) = \langle I(\Gamma)_{\le m-1}, \Phi(\Gamma)_m \rangle$, and hence is generated in degrees $\le m$. The core technique uses the geometry of the split variety $Split_m(\mathbb{P}^n)$ to prove that $\Phi(\Gamma)_m$ spans $I(\Gamma)_m$, and then extends to degree $m+1$ via $S_1 \Phi(\Gamma)_m$ together with Castelnuovo–Mumford regularity, yielding a constructive description of the whole ideal. For the special case $m=2$ (i.e. $d \le 2n$) the result rederives Saint-Donat’s quadratic-generation of $I(\Gamma)$ of rank $2$, thereby strengthening Treger’s bound and clarifying the role of completely decomposable forms in the defining equations of point sets. Overall, the paper provides a new, geometric proof framework and explicit generating sets for the defining equations of finite sets in general position, with concrete combinatorial links to partitions of $\Gamma$ and to the degree of the split variety.

Abstract

The study of the defining equations of a finite set $Γ\subset \mathbb{P}^n$ 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 $I(Γ)$ is generated by forms of degree $\leq \lceil \frac{|Γ|}{n}\rceil$. 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 $I(Γ)$ is generated by the union of $I(Γ)_{\leq \lceil \frac{|Γ|}{n}\rceil -1}$ and the set of all completely decomposable forms of degree $\lceil \frac{|Γ|}{n}\rceil$ in $I(Γ)$. In particular, it holds that if $d \leq 2n$ then $I(Γ)$ is generated by quadratic equations of rank $2$. This reproves Saint-Donat's results in \cite{SD1} and \cite{SD2}.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.2
• Theorem 2.1
• Proposition 2.2
• proof
• Lemma 2.4
• proof
• Theorem 3.1
• Lemma 3.2
• proof