The Artinian Point Star Configuration Quotient and the Strong Lefschetz Property

J. Korean Math. Soc. Published online 2019 Mar 04

Young-Rock Kim, and Yong-Su Shin
Hankuk University of Foreign Studies, Sungshin Women University

Abstract : It has been little known when an Artinian points quotient has the SLP. In this paper, we find an Artinian points star configuration quotient having the SLP. If $\mathbb X$ is a star-configuration in $\mathbb P^2$ defined by $s$-forms ($a$-quadratic forms and $(s-a)$-linear forms) and $\mathbb Y$ is a star-configuration defined by $t$-forms ($(t-b)$-quadratic forms and $b$-linear forms) for $b=\deg(\mathbb X)$ or $\deg(\mathbb X)-1$, then an Artinian ring $R/(I_{\mathbb X}+I_{\mathbb Y})$ has the SLP. Moreover, if $\mathbb X$ and $\mathbb Y$ are linear star configuration in $\mathbb P^n$ of type $s$ and $t$ with $s\ge t$ and $t=n$ or $n+1$, respectively, then an Artinian ring $R/(I_{\mathbb X}+I_{\mathbb Y})$ has the SLP.

Keywords : Hilbert functions, Generic Hilbert functions, Star configurations, Linear star connfigurations