On completely decomposable defining equations of points in general position in $\mathbb{P}^n$
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| Format: | Preprint |
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2020
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| _version_ | 1866916561589108736 |
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| author | Jung, Jaeheun Park, Euisung |
| author_facet | Jung, Jaeheun Park, Euisung |
| contents | 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}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2007_06893 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On completely decomposable defining equations of points in general position in $\mathbb{P}^n$ Jung, Jaeheun Park, Euisung Algebraic Geometry Commutative Algebra 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}. |
| title | On completely decomposable defining equations of points in general position in $\mathbb{P}^n$ |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/2007.06893 |