The Artinian point star configuration quotient and the strong Lefschetz property
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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.
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.
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