On completely decomposable defining equations of finite sets in Pn

Abstract

Let $I(\Gamma )$ be the homogeneous ideal of a finite set Γ in ${\mathbb{P}}^n$ of d points in linearly general position. In 1972, Saint-Donat proved that if $d\le 2n$ then $I(\Gamma )$ is generated by completely decomposable quadratic polynomials. Later, R. Treger generalized this result by showing that $I(\Gamma )$ is generated by forms of degree $\le m$ if $d\le \mathit{mn}$ for some $m\ge 2$.

This paper aims to generalize Saint-Donat’s work in a different direction by proving that the degree m piece of $I(\Gamma )$ can be generated by completely decomposable forms of degree m if and only if $d\le \mathit{mn}$.

Communicated by Daniel Erman

KEYWORDS:

• Completely decomposable form
• defining equations of a finite set
• linearly general position

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14N05
• 51N35
• 14C17

Additional information

Funding

The second named author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (no. 2022R1A2C1002784). We are grateful to the anonymous referee for careful reading and valuable suggestions.