Journal of the
Korean Mathematical Society JKMS

In this paper, we study an Artinian point-configuration quotient having the SLP. We show that an Artinian quotient of points in $\mathbb P^n$ has the SLP when the union of two sets of points has a specific Hilbert function. As an application, we prove that an Artinian linear star configuration quotient $R/(I_\mathbb X+I_\mathbb Y)$ has the SLP if $\mathbb X$ and $\mathbb Y$ are linear star-configurations in $\mathbb P^2$ of type $s$ and $t$ for $s\ge (\frac{t}{2})-1$ and $t\ge 3$. We also show that an Artinian $\mathbb k$-configuration quotient $R/(I_\mathbb X+I_\mathbb Y)$ has the SLP if $\mathbb X$ is a $\mathbb k$-configuration of type $(1,2)$ or $(1,2,3)$ in $\mathbb P^2$, and $\mathbb X\cup\mathbb Y$ is a basic configuration in $\mathbb P^2$.

Keywords: Hilbert functions, star-configurations, linear star-configura\-tions, the weak-Lefschetz property, the strong-Lefschetz property

MSC numbers: Primary 13A02; Secondary 16W50