论文标题
在$ \ mathbb {p}^n $中完全可分解的定义方程
On completely decomposable defining equations of points in general position in $\mathbb{P}^n$
论文作者
论文摘要
有限集的定义方程式$γ\ subset \ mathbb {p}^n $线性通用位置的研究已被积极吸引，因为它在理解算术上的Cohen-Macaulay品种的定义方程中起着重要作用。在\ cite {t}中，R。Treger证明了$ i（γ）$是由$ \ leq \ lceil \ frac \ frac {|γ|} {n} {n} \ rceil $生成的。从那时起，Treger的结果已被扩展并改善了几篇论文。 本文的目的是从新的角度谴责和改善上述Treger的结果。本文我们的主要结果表明，$ i（γ）$是由$ i（γ）的联合生成的， $ i（γ）$。特别是，如果$ d \ leq 2n $，则$ i（γ）$由二次方程式等级$ 2 $生成。这将在\ cite {sd1}和\ cite {sd2}中依靠圣多纳特的结果。
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}.
