The Minimal Free Resolution of A Star-Configuration in $\mathbb{P}^n$

TL;DR

This paper determines the minimal free resolution of ideals defining star-configurations in projective space, generalizing previous results and proving these configurations are arithmetically Cohen-Macaulay, with applications to constructing Artinian rings with the weak Lefschetz property.

Contribution

It extends the minimal free resolution results of star-configurations to all types $(r,s)$ in $ ext{P}^n$ and proves their Cohen-Macaulay property, with applications to Artinian rings.

Findings

Resolved minimal free resolutions for star-configurations of any type $(r,s)$.

Proved star-configurations are arithmetically Cohen-Macaulay.

Constructed Artinian rings with the weak Lefschetz property.

Abstract

We find the minimal free resolution of the ideal of a star-configuration in $\mathbb{P}^n$ of type $(r,s)$ defined by general forms in $R=\Bbbk[x_0,x_1,\dots,x_n]$. This generalises the results of \cite{AS:1,GHM} from a specific value of $r=2$ to any value of $1\le r\le n$. Moreover, we show that any star-configuration in $\mathbb{P}^n$ is arithmetically Cohen-Macaulay. As an application, we construct a few of graded Artinian rings, which have the weak Lefschetz property, using the sum of two ideals of star-configurations in $\mathbb{P}^n$.