A graded minimal free resolution of the $m$-th order symbolic power of a star configuration in $\mathbb P^n$
J. Korean Math. Soc. 2021 Vol. 58, No. 2, 283-308
https://doi.org/10.4134/JKMS.j190739
Published online February 4, 2021
Printed March 1, 2021
Jung Pil Park, Yong-Su Shin
Seoul National University; KIAS
Abstract : 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 $m\ge 5$.
Keywords : Symbolic powers, regular powers, star configurations, a graded minimal free resolution
MSC numbers : 13A15, 14M05
Supported by : This research was supported by the Basic Science Research Program of the NRF (Korea) under grant No.2019R1F1A1056934
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