A graded minimal free resolution of the $m$-th order symbolic power of a star configuration in $\P^n$

In \cite{S:3} the author finds a graded minimal free resolution of the $2$-nd order symbolic power of a star configuration in $\P^n$ of any codimension $r$. In this paper, we find that of any $m$-th order symbolic power of a star configuration in $\P^n$ of codimension $2$, which generalizes the result of Galetto, Geramita, Shin, and Van Tuyl in \cite[Theorem 5.3]{GGSV:1}. Furthermore, we extend it to the $m$-th order symbolic power of a star configuration in $\P^n$ of any codimension $r$ for $m=3,4$, which also generalizes the result of Biermann et al. in \cite[Corollaries 4.6 and 5.7]{BDGMNORS}. We also suggest how to find a graded minimal free resolution of the $m$-th order symbolic power of a star configuration in $\P^n$ of any codimension $r$ for $m\ge 5$.