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

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 $R=\mathbb{k}[x_0,x_1,\dots ,x_n]$. 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 ${\mathbb{P}}^n$.