The Artinian point star configuration quotient and the strong Lefschetz property
J. Korean Math. Soc. 2019 Vol. 56, No. 3, 645-667
https://doi.org/10.4134/JKMS.j180258
Published online 2019 May 01
Young-Rock Kim, Yong-Su Shin
Hankuk University of Foreign Studies; Sungshin Women's University, KIAS
Abstract : It has been little known when an Artinian point quotient has the strong Lefschetz property. In this paper, we find the Artinian point star configuration quotient having the strong Lefschetz property. We prove that if $\mathbb X$ is a star configuration in $\mathbb P^2$ of type $s$ defined by forms ($a$-quadratic forms and $(s-a)$-linear forms) and $\mathbb Y$ is a star configuration in $\mathbb P^2$ of type $t$ defined by forms ($b$-quadratic forms and $(t-b)$-linear forms) for $b=\deg(\mathbb X)$ or $\deg(\mathbb X)-1$, then the Artinian ring $R/(I_\mathbb X+I_\mathbb Y)$ has the strong Lefschetz property. We also show that if $\mathbb X$ is a set of $(n+1)$-general points in $\mathbb P^n$, then the Artinian quotient $A$ of a coordinate ring of $\mathbb X$ has the strong Lefschetz property.
Keywords : Hilbert functions, generic Hilbert functions, star configurations, linear star configurations
MSC numbers : Primary 13A02; Secondary 16W50
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