J. Korean Math. Soc. 2019; 56(3): 645-667
Online first article March 4, 2019 Printed May 1, 2019
https://doi.org/10.4134/JKMS.j180258
Copyright © The Korean Mathematical Society.
Young-Rock Kim, Yong-Su Shin
Hankuk University of Foreign Studies; Sungshin Women's University, KIAS
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
Supported by: The first author was supported by Hankuk University of Foreign Studies Research Fund.
The second author was supported by the Basic Science Research Program of the NRF (Korea) under grant No.2016R1D1A1B03931683/3.
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